Unit 9 Radical Expressions and Equations Introduction The real numbers are made up of many types of numbers. The simplest set of numbers is called the Natural Numbers. These numbers be can written as the list given below. Natural Numbers = { 1, 2, 3, 4, … } There is also a set of numbers called the Integers. These numbers can also be shown as a list. Integers = { 0, 1, -1, 2, -2, 3, -3, … } Another set of numbers is called the Rational Numbers because they can be represented as fractions or ratios. This set of numbers can also be described as all the terminating or repeating decimal numbers. Some examples of values from this set are given below. = 1.25 This is a terminating decimal value. ( Terminating means it stops. ) = 1.33333… = 1.3 This is a repeating decimal value. = .18181818… = .18 This is also a repeating decimal value.
Transcript
Unit 9 Radical Expressions and Equations
IntroductionThe real numbers are made up of many types of numbers. The simplest set
of numbers is called the Natural Numbers. These numbers be can written as the list given below.
Natural Numbers = { 1, 2, 3, 4, … }
There is also a set of numbers called the Integers. These numbers can also be shown as a list.
Integers = { 0, 1, -1, 2, -2, 3, -3, … }
Another set of numbers is called the Rational Numbers because they can be represented as fractions or ratios. This set of numbers can also be described as all the terminating orrepeating decimal numbers. Some examples of values from this set are given below.
= 1.25 This is a terminating decimal value. ( Terminating means it stops. )
= 1.33333… = 1.3 This is a repeating decimal value.
= .18181818… = .18 This is also a repeating decimal value.
= .142857142857… = .142857 Again we have a repeating decimal value.
The set of numbers we are going to study in this unit is called the Irrational Numbers.This set of numbers is the opposite of the Rational Numbers. The Irrational numbersare decimal values that do not terminate and do not repeat. Below are some examplesof these numbers.
= 3.1415926535897932384626 … The value of pi is irrational, it will never
stop and never form a pattern in order to repeat.
Any square root that does not terminate will be an irrational number.
= 1.41421356 … This value will never terminate and never form a repeating
pattern. So the , like most square roots, is irrational.
.
Unit 9 Vocabulary and ConceptsIndex The index of a radical tells us the number of times a factor must be used
to generate the radicand.
Radical A radical is the symbol which tells us to reverse a square. The understood index here is 2.
Radicand The radicand is the value or expression inside the radical.
Key Concepts for Exponents and Polynomials
Simplifying Radicals To simplify radicals means to change any perfect squares into
their roots. If some factors of the radicand are perfect
squares and some are not, then we split the radical into
two roots - one with the perfect squares, and one with the
factors that are not perfect squares.
Multiplying Radicals To multiply radicals or square roots we multiply the
radicands
and simplify the radical.
Adding/Subtracting Radicals To add/subtract radicals we add or subtract
the
coefficients of the radicals. This can only be
done
when the radicands and indices are the same.
Dividing Radicals To divide radicals or square roots we divide out the
common
factors of the radicands and simplify the radicals and
rationalize
the denominator where necessary.
Rationalizing the Denominator To rationalize a denominator we multiply
the
fraction by the denominator over itself.
Objective The student will multiply radicals.
In order to work with radicals we should first make sure we can use the vocabulary associated with these numbers. The expression below shows us a radical value and the terms we use for it.
The radical is the symbol in the example. This symbol means we are reversing a square, cube, or 4th power.The radicand is the number or expression inside the radical.The index tells us how many times a factor must be used to multiply to the number or expression in the radical to produce the radicand.
Below are some examples of radical expressions and their meanings.
Find the number that times itself equals twenty-five. This evaluates to 5 because 52 = (5)(5) = 25.
Find the number that used as a factor three times equals 64. This evaluates
to 4 because 43 = (4)(4)(4) = 64.
Find the number that used as a factor four times equals 81. This evaluates
to 3 because 34 = (3)(3)(3)(3) = 81.
In these examples we must emphasize that a radical without an index is understood to be a square root. This means that the understood index is ‘2’. In Algebra I we will be primarily concerned with the square roots.
When we work with radicals or square roots we must understand the difference betweenthe exact value of a number and an approximation of a number. The square root of ‘2’, when written with a radical is exact; when written as a decimal, it is rounded off and so is an approximation.
Exact Approximation 1.41421
Unit 9 Section 1
Index
Radical
Radicand
It will not matter how many decimal places we use with the value on the right it will always be an approximation.
To find the exact answers we will need to be able to perform operations with radicals or square roots. We will also at times have variables and expressions inside the radical. When there are expressions inside a radical we cannot find an approximation and we must be able to do operations with the radicals themselves. The first operation we need to explore is multiplication of square roots. We will examine some problems to develop a method for multiplication.
Example A . This problem can be done with two different methods.
Method 1 Method 2 . .
2 . 3 6 6
Example B . This problem can be done with two different methods.
Method 1 Method 2 . .
5 . 4 20 20
Example C . This problem can be done with two different methods.
Method 1 Method 2 . .
2 . 5 10 10
From examples A through C we can see that we can multiply the radicands, the numbers in the radicals, first and then take the square roots afterwards if we want. There is one restriction that we need to know. We can only multiply radicals that have the same index. We will normally only be using square roots.
Below are some examples of multiplying square roots and simplifying when possible. Example A Example B Example C Example D
. . . . .
VIDEO LINK: Khan Academy Multiply and Simplify a Radical Expression 1
Exercises Unit 9 Section 1Find the square roots. If there is no real answer then write “no real number”.
1. 2. 3. 4. –Estimate the square roots to the nearest tenth. Check your answers by multiplyingyour estimate times itself. Show your work.
5. 6. 7. 8. Evaluate the following. Show your work.
9. Evaluate 3 – 4 with x = 14
10. Evaluate - + 3 with x = 11
11. Given f(x) = Find f(7) and f(21), show your work.
Multiply the following and simplify where possible. 12. . 13. . 14. . 15. . 16. . 17. . 18. . 19. . 20. . 21. . 22. . . 23. . 24. . 25. . . 26. .
Since (a2)(a2) = a4
the square root of a4 must be a2.
Since (6x)(6x) = 36x2
the square root of 36x2 must be 6x.
Since there are no perfect squares in the radicand we cannot simplify this answer.
Since (15xy3)(15xy3) is equal to 225x2y6 the square root of 225x2y6 must be 15xy3.
Simplifying square roots requires that we recognize perfect squares. The list below contains all the perfect squares from 1 to 225. These are the squares of the natural numbers from 1 to 15.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225Variable expressions can also be perfect squares. The list below contains examplesof these expressions.
x2 = (x)(x), x4 = (x2)(x2), x6 = (x3)(x3), x8 = (x4)(x4), x10 = (x5)(x5) Any variable with an even exponent is a perfect square.
In the last section we were asked to multiply and simplify square roots. The radicals that we were asked to simplify usually contained only perfect squares. These would be problems similar to the ones shown below.
There were also a few problems that we could not simplify such as those
given below.
These problems have no perfect squares. There are problems that have both types
of factors in the radicand. When we have factors that are perfect squares, then
simplifying the radical requires that we simplify the radical as much as possible. This
will mean splitting the radical into factors of perfect and non-perfect squares. The examples below show us this process.
Example A Simplify
Unit 9 Section 2
The value 75 is not a perfect square. However, 75 can be factored into 25 . 3,
and 25 is a perfect square. This means we can write the 75 in factored form.
We factored the radicand. . Since 25 . 3 = 75 we can write this as a product of two
radicals. 5 . The square root of 25 is 5 so we simplify the first square
root. 5 We don’t have to write the dot for multiplying so this is the best form.
Example B Simplify The value 44 is not a perfect square. However, 44 can be factored into 4 .
11,and 4 is a perfect square. This means we can write the 44 in factored form.
We factored the radicand. . Since 4 . 11 = 44 we can write this as a product of two radicals. 2 . The square root of 4 is 2 so we simplify the first square root. 2 We don’t have to write the dot for multiplying so this is the best
form.
In these first two examples we put the perfect square in the first radical each time. This makes our simplifying a little easier. When we split the radical into the product of two radicals we will always put the perfect squares into the first radical. We should always find the largest perfect square that is a factor as well.
Example C Simplify The value 128 is not a perfect square. However, 128 can be factored into 64 .
2,and 64 is the largest perfect square. So we can write the 128 in factored
form.
We factored the radicand. . Since 64 . 2 = 128 we can write this as a product of two radicals. 8 . The square root of 64 is 8 so we simplify the first square root.
8 We don’t have to write the dot for multiplying so this is the best form.
Example D Simplify The value 8x7 is not a perfect square. But, 8x7 can be factored into 4x6 . 2x,and 4x6 is a perfect square. This means we can write the 8x7 in factored form.
We factored the radicand.
. Since 4x6 . 2x = 8x7 we can write this as a product of two radicals.
2x3 . The square root of 4x6 is 2x3 so we simplify the first square root.
2x3 We don’t have to write the dot for multiplying so this is the best form.
Example E Simplify The value 98xy2z5 is not a perfect square. However, 98xy2z5 can be
factored into 49y2z4 . 2xz, and 49y2z4 is a perfect square. This means we can
write the 98xy2z5 in factored form.
We factored the radicand. . Since 49y2z4 . 2xz = 98xy2z5 this can be the product of two radicals.
7yz2 . The square root of 25 is 5 so we simplify the first square root.
7yz2 We don’t have to write the dot for multiplying so this is the best form.
VIDEO LINK: Khan Academy Simplifying Square Roots
Exercises Unit 9 Section 2Multiply the following and simplify where possible.
Objective The student will add and subtract square roots with like radicands.
The next operations we need to perform are addition of radicals or square roots. Since subtraction is the addition of the opposite we can do both addition and subtraction atthe same time. To understand how addition of square roots is done we need to review how we combine like terms. The examples below should help us remember.
Example A Example B Example C
x + x + x + x 4a + 3a - a 2x + 3y + z 4x 6a cannot be combined
In these examples we see that as long as the variable is the same we can add orsubtract the coefficients of the like terms. We can also see this process when weuse numeric values.
Example A Example B
1(7) + 1(7) + 1(7) + 1(7) 4(5) + 3(5) – 1(5)
Unit 9 Section 3
4(7) 6(5) 28 30
In Example A we were adding four sevens together but we could also have been addingthe coefficients and got 4 times 7. In Example B again we added/subtracted the coefficients to get 6 times 5. It would not really matter what the value in the parentheses is, the process would be the same. As long as the numbers in the parentheses are the same we can add the coefficients. The examples below show us how to add square roots.
Example A Example B Example C
+ + + + 2 3 Cannot be added
Example D Example E Example F
4 + 5 – + 5 – 9 -4 0
Example G Example H Example I
7 + x – y 11 – 8 (x – y) 9
VIDEO LINK: Khan Academy Adding and Simplifying Radicals
Exercises Unit 9 Section 3
Multiply the following and simplify where possible. 1. . . 2. . 3. . 4. . 5. . 6. .
Simplify each square roots as much as possible. Show how you split the radicals into a product for each problem.
Simplify then multiply the resulting monomials. Show your work.
53. (a3b2)4(ab)3 54. (-2a4b)3(-ab3)2
Objective The student will add and subtract square roots.
We know that radicals can only be added or subtracted when the radicals have the sameradicands. There are problems that initially seem to have different radicands. Some of these problems can be done if we simplify one or both of the radicands. The examplesbelow show us how to simplify and add radicals.
Example A + The radicands in this example appear to be different. However,
the can be simplified.
+ . + 2 +
3
Example B +
Unit 9 Section 4
Since 20 can be factored into 4 times 5 we can write it as a product of two radicals.
After simplifying , the radicands are the same so we can add the radicals.
The radicands in this example appear to be different. However, both
of the radicals can be simplified.
+ . + .
2 + 3 5
Example C + + The radicands in this example appear to be different. However,
allof the radicals can be simplified.
+ – . + . – .
5 + 3 – 2 6
Example D + The radicands in this example appear to be different. However, allof the radicals can be simplified.
+ . + .
10 + 2a
Exercises Unit 9 Section 4
Multiply the following and simplify where possible. 1. . . 2. . 3. . 4. . 5. . 6. .
Simplify each square root as much as possible. Show how you split the radicals into a product for each problem.
7. 8. 9. 10. 11. 12. 13. 14. 15.
Answer the following given the expression
Since 28 is equal to 4 times 7 and 63 has factors of 9 and 7 so both can be simplified.
After simplifying, the radicands are the
Since 125 = 25 . 5, 45 = 9 . 5 and (10)(2) = 20 = 4 . 5 we can simplify all three.
After simplifying, the radicands are the same
Since 200a = 100 . 2a and 8a3 = 4a2 . 2a we can simplify both radicals.
After simplifying, the radicands are the same
16. What is the radicand?17. What is the index?
Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
Divide and list your answer as both a fraction and with negative exponents.
51. 52. 53.
Objective The student will divide square roots.
We can now simplify, add, subtract and multiply square roots or radicals. The operationwe still need to study is division. We will use the examples below to investigate how our method for division will work.
Example A Simplify
There are two methods we can use to do this problem.Method 1 Method 2
2 2Both methods give us the same answer ‘2’. In Method 1 we simplify each of the two radicals or square roots separately and then divide. In Method 2we divide the radicands and only then simplify the square root.
Example B Simplify
Unit 9 Section 5
There are two methods we can use to do this problem.Method 1 Method 2
10 10Both methods give us the same answer ‘10’. In Method 1 we simplify each of the two radicals or square roots separately and then divide. In Method 2we divide the radicands and only then simplify the square root.
While both methods work and will produce the same answer Method 2, where we divide the radicands first, has some advantages. Consider the problem below.
Example C Simplify
There are two methods we can use to do this problem.Method 1 Method 2
5
5
Both methods give us the same answer ‘50’. In Method 1 we simplify each
of the two radicals, or square roots, separately and then factor and divide. In Method 2 we divide the radicands and only then simplify the square root.
In order to make our work simpler we should try to divide the radicands whenever
possible. The examples that follow show us how this process can be used.
Example A Example B Example C
5 . 7
In Example A and C, the division leaves us with a perfect square to simplify whilein Example B, the quotient is not a perfect square but can still be simplified.
Example D Example E Example F
3xz2
VIDEO LINK: Khan Academy Multiply and Simplify a Radical Expression 2
Exercises Unit 9 Section 5
Multiply the following and simplify where possible. 1. . . 2. . 3. .
Simplify each square root as much as possible. Show how you split the radicals into a product for each problem.
4. 5. 6. Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
7. 4 + 9 8. – 11 9. + 10. 3 – 5 11. w + 2 12. +
We need to recall that if we divide monomials with exponents we subtract the powers.
After we divide if we are not left with monomials that simplify then we are done.
After we divide sometimes the monomials will simplify completely.
Divide the radicals and simplify where possible. Show your work.
19. 20. 21.
22. 23. 24.
25. 26. 27.
28. 29. 30.
31. 32. 33.
34. 35. 36.
Solve the following by completing the square. Round your answers to the nearest hundredth where needed. Show your work.
37. x2 + 12x + 28 = 0 38. 2x2 + 12x = 8
Use the Quadratic Formula to solve the following. Show your work. Do NOT find decimal approximations leave square roots in your answers where needed.
39. x2 - 23x + 76 = 0 40. 4a2 – 31a = 8
Objective The student will divide square roots and rationalize denominators.
In all the problems we saw in Section 6 the division did not leave any factors in the denominator. The problems we will see now could leave radicals in the numerator and denominator. The best form for expressions with radicals requires that we simplify the expression and not leave radicals in the denominator. The process for removing a radical from the denominator is called rationalizing the denominator. There are three principles we must be able to use in this process. These principles are demonstrated below.
Any number divided by itself is one.
Example A Example B Example C
= 1 = 1 = 1
This is something we have known for a long time. We need to understand that
this principle also applies to square roots or radicals.
Any square root times itself equals the radicand.
Example A Example B Example C
Unit 9 Section 6
. = = 5 . = = 7 . = =
This can be restated as any square root times itself equals the value in the radical.
One times any number equals the original number.
Example A Example B Example C
7 . 1= 7 . 1 = . 1 =
Rationalizing the denominator of a fractional expression requires understanding all three principles. A given number can be written in many forms. For instance the value ‘2’ can be written in many ways.
2, , 2.0, , , , …
Some of these forms are easier to work with than others but they are all equal to ‘2’.The preferred form of a fraction with a radical in it is with only one radical in the numerator. The next set of examples will demonstrate the process of rationalizing the denominator.
Example A Rationalize the denominator of
.
We cannot change the value of the expression only the form as we go through this process.We are multiplying by 1 so the value will not change.We are using the denominator over itself as our When we multiply fractions we multiply straight across.The final denominator is the radicand of the square root in the bottom of the fraction.
Example B Rationalize the denominator of
.
Example C Rationalize the denominator of
.
Example D Rationalize the denominator of
.
We cannot change the value of the expression only the form as we go through this process.We are multiplying by 1 so the value will not change.We are using the denominator over itself as our When we multiply fractions we multiply straight across.The final denominator is the radicand of the square root in the bottom of the fraction.
We cannot change the value of the expression only the form as we go through this process.We are multiplying by 1 so the value will not change.We are using the denominator over itself as our The final denominator is the radicand of the square root in the bottom of the fraction.We factored the ‘8’ so we could divide out the 2’s. There will be times when the denominator divides out.
We cannot change the value of the expression only the form as we go through this process.We are multiplying by 1 so the value will not change.We are using the denominator over itself as our The final denominator is the radicand of the square root in the bottom of the fraction.
Example E Rationalize the denominator of
.
Fractions inside of radicals can also be simplified with this process. One of the keys to this process is to change the fraction into a division of square roots. The examples below illustrate this relationship.
Example F Find the
Method 1 Method 2
Since
we can write the radical as
We factored the ‘10’ and ‘6’ so we could divide out the 2’s.
We cannot change the value of the expression only the form as we go through this process.We are multiplying by 1 so the value will not change.We are using the denominator over itself as our The final denominator is the radicand of the square root in the bottom of the fraction.We factored the ‘6’ so we could divide out the 3’s.
Because dividing square roots is done by dividing radicands we can reverse this and make a radicand with division into the dividing of square
The final answer is The final answer is
This example tells us the best way to simplify the square root of a fraction is to make it into the division of two square roots. This is Method 2 above.
Example G Find the
Write the fraction as the division of square roots.
We simplified the square roots separately.
Example H Find the
We write the fraction as the division of square roots.
.
Example I Find the
First we divide out the common factor of ‘b’.
We write the fraction as the division of square roots.
We can simplify the numerator but now we need torationalize the denominator.We are multiplying by 1 so the value will not change.We are using the denominator over itself as our
We can simplify the numerator but now we need torationalize the denominator.
The final denominator is the radicand of the square root in the bottom of the fraction.
.
We may also be asked to divide two square roots and rationalize the denominator where necessary.
Example J Divide and rationalize the denominator of
=
We simplified the numerator since it was a perfect square.
.
VDEO LINK: Youtube: Rationalizing the Denominator
Exercises Unit 9 Section 6Simplify each square root as much as possible. Show how you split the radicals into a product for each problem.
1. 2. 3. Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
Divide the radicals and simplify where possible. Show your work.
We are multiplying by 1 so the value will not change.We are using the denominator over itself as our The final denominator is the radicand of the square root in the bottom of the fraction.
We cannot change the value of the expression only the form as we go through this process.
We are multiplying by 1 so the value will not change.We are using the denominator over itself as our The final denominator is the radicand of the square root in the bottom of the fraction.
First we factor the radicands and divide out the common factors.
Divide the radicals and simplify where possible. Rationalize denominators wherenecessary. Show your work.
19. 20. 21.
22. 23. 24.
25. 26. 27.
28. 29. 30.
31. 32. 33.
Simplify the following. Remember to rationalize the denominators as needed.
34. 35. 36.
37. 38. 39.
40. 41. 42.
43. 44.
Graph the following quadratic equations using substitution. Show your tables and graphs.If you completed a square show the work converting the equation’s form.
45. y = (x – 2)2 – 5 46. y = x2 + 6x + 2
Objective The student will solve contextual problems using square roots.
There are many kinds of problems that result in, or use, square roots. Geometric problems are among the best examples.
Example A Find an expression for the area of the rectangle below. 3 +
A = l . w First we write the formula for the area.
A = ( )(3 + ) Second we substitute for ‘l’ and ‘w’.
A = ( )(3 + ) Now we use the Distributive Property.
A = 3 + Finally we simplify. A = 3 + This is our expression.
Example B Given the rectangle below find the length of the diagonal. 13 m
x 9 cm
a2 + b2 = c2
92 + 132 = x2 81 + 169 = x2
250 = x2
= . = x
Unit 9 Section 7
Geometry ReviewTo find the length of the diagonal we must use the Pythagorean Theorem. This theorem states that in a right triangle the sides have the relationship given by the diagram and equation below.
ca2 + b2 = c2 a
b
5 = x
Example C A rectangle has a length of 4 cm and a width of 7 cm find the perimeter. When we have a question about a rectangle often the best way to start the problem is to draw a rectangle and label the diagram.
4 7
The perimeter for any figure is found by adding up all the sides. So for the figure above we should create the equation below.P = 4 + 4 + 7 + 7 We can add some terms.P = 8 + 14 Then simplify the radicals.P = 8 . . + 14 . .
P = 8 . 5 . + 14 . 2 .P = 40 + 28P = 68
Example D A rectangle has a length of 3x and a width of x. If the area of the rectangle is 49 m2 find ‘x’ and the dimensions of the figure.
3x
x First draw and label the figure.
A = l . w We write the formula for the area. 49 = (3x)(x) Next we substitute for the variables. 49 = 3x2 We multiply the monomials. 3 3 Use the Division Property of Equality. = x2 Simplify the equation.
= Take square roots on both sides.
= x Simplify the equation.
. = x Rationalize the denominator.
= x And we have our answer.
Example E A rectangle has a length of + 3 and a width of – 3.
Find the expression for the area. + 3
– 3 We draw and label the rectangle.
A = l . w We write the formula for the area.A = ( + 3 )( – 3 ) Next we substitute.The two binomials in the product are actually sum and difference
binomials. We can use either FOIL or our short cut and multiply the binomials.
A = ( + 3 )( – 3 ) We use the sum and difference multiplication.
A = ( )2 – (3 )2 The result of sum & difference multiplication
A = ( . ) – (3 . 3 ) We expand the squares & multiply the radicals.
A = – 9A = x – 9y This is the answer after simplifying the
radicals.
In this example there is one step that we need to look at more closely. In the step starting from A = ( )2 – (3 )2 we had to simplify the squares. This can be done by expanding the squares and multiply the radicals. There is another way to perform this operation. The examples below will help us understand this method.
Example A ( )2 = . = = so ( )2 = Example B ( )2 = . = = so ( )2 = Example B ( )2 = . = = so ( )2 =
From these three examples our conclusion should be:
If we raise a square root to the 2nd power the answer will be the radicand.
Exercises Unit 9 Section 7
Solve the following contextual problems. Show your equations and your work to find the solutions. All answers must be exact and must be simplified as much as possible.
1. Given the rectangle below as labeled find the area.
cm
cm
2. Given the right triangle below as labeled find the length of the diagonal. m
m x
3. Given the rectangle below as labeled with an area of 20 in2 find the value of ‘x’ and the dimensions.
in
in
4. Given the rectangle below as labeled find the area of the figure. 7 + ft
7 – ft
5. Given the rectangle below with the length and diagonal as labeled find the width.
15 cm
19 cm x
6. Given the circle below has an area of cm2 find the radius. (A = )
x
7. Given the right triangle as labeled below find the area of the figure. m
m
8. Given the rectangle as labeled below find the perimeter. cm
cm
9. Given the square below has an area of in2 find the side. (A = s2)
x
10. If a triangle has sides of cm, cm, and cm find the perimeter.
11. If a rectangle has length of 8x, a width of 2x, and an area of 81 m2, find the value of ‘x’ and the dimensions.
12. If a rectangle has sides of a + and find the expression for the area.
13. Given a right triangle as shown below find the hypotenuse.
mx
m
14. If a rectangle has a width of cm and a diagonal of 16 cm find the length and the perimeter.
15. If a circle has a radius of find the expression for its area.
16. Given the trapezoid below as labeled find the area of the figure. (A = )
in
in
in
17. If a square has a side of 2x and an area of 169 cm2 find the value of ‘x’ and the length of the side.
18. Mike has built a backdrop panel for a school play. The length of the panel is ft. and the height is ft. He needs to put a diagonal stabilizer on the panel. What will the measurement of the diagonal need to be?
19. A solar panel needs to have an area of 256 in2. The panel will be a rectangle. The length should be nine times its width. Find the dimensions of the panel.
20. Given the isosceles triangle below as labeled find the altitude of the triangle.
”
”
x
Objective The student will solve literal equations using square roots.
Solving literal equations means that we are solving an equation for a particular variable.This is a skill we learned in Unit 2. Solving a literal equation will result in the isolated variable being equal to an expression not just a real number. The two examples below illustrate this difference.
Example A Example Bx2 − 7x + 12 = 0 ab + c = d solve for b
(x − 3)(x − 4) = 0 -c -c x = 3, 4 ab = d − c
a a b = d − c
a
In Example A we solved the equation and the variable 'x' was equal to 3 or 4, real numbers. In Example B we isolated the variable 'b' however the result is an expression.Example B shows us the process for solving a literal equation.
When we studied this skill in Unit 2 we used it to isolate variables that had the understood exponent of 1. We have progressed beyond this point and we can nowhave quadratic equations or even equations with radicals. The examples that followshow us how to isolate variables from quadratic and radical equations.
Example A Given the formula A = 6s2 isolate s. (The surface area of a cube)
We use the Division Property of Equality.
We need to rationalize the denominator.
Unit 9 Section 8
We take the square root on both sides in order to reverse the square of the 's2'
term.
.
This is the final answer.
Example B Given y = (x - b)2 - c isolate x.
We use the Addition Property of Equality.
The square is now isolated.
Example C Given isolate z.
We use the Division Property of Equality.
We use the Subtraction Property of Equality.
We take square roots on both sides. When the square is isolated the next step is taking the square root.We use the Addition Property of
Since the square root is now isolated we can reverse the root and square both sides.
Exercises Unit 9 Section 8
Isolate the asked for variable in each formula or equation. Show all your work and be prepared to explain all your steps. Rationalize all denominators.
1. A = s2 isolate s 2. A = isolate r
3. a2 + b2 = c2 isolate c 4. a2 + b2 = c2 isolate a
5. V = isolate r 6. w = z2 + 1 isolate z
7. z = (a + 2)2 − 1 isolate a 8. z = 3y2 isolate y
9. 4m = 2(w − z)2 isolate w 10. isolate x
11. isolate b 12. isolate z
Find the square roots. If there is no real answer then write “no real number”.
13. 14. 15. -
Estimate the square roots to the nearest tenth. Check your answers by multiplyingyour estimate times itself. Show your work.
16. 17. Evaluate the following. Show your work. If the answer is non-real state "no real answer".
18. Evaluate y = – 1 with x = 11719. Evaluate y = + with x = 11
20. Evaluate y = with x = 10
21. Evaluate y = - with x = 4422. Given f(x) = Find f(9), f(1), and f(10) show your
work.23. Given g(x) = Find g(9), g(30), and g(1) show your
work.
Multiply the following and simplify where possible. 24. . 25. . 26. . 27. . 28. . 29. . 30. . 31. . . 32. .
Simplify the each square root as much as possible. Show how you split the radicals into a product for each problem.
33. 34. 35. 36. 37. 38.
Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
39. + 40. 7 – 7 41. – 1842. + 43. + 7 44. 3b – 4b
Simplify, then add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
45. + 46. – 47. + 48. + 49. – + 50. +
Divide the radicals and simplify where possible. Show your work.
51. 52. 53.
54. 55. 56.
Divide the radicals and simplify where possible. Rationalize denominators where
necessary. Show your work.
57. 58. 59.
60. 61. 62.
Simplify the following. Remember to rationalize the denominators as needed.
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