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CONFIDENTIAL 1
Algebra1Algebra1
RadicalRadicalExpressionsExpressions
CONFIDENTIAL 2
Warm UpWarm Up
1) {(-3, 16), (-2, 8), (0, 2), (1, 1), (3, 0.25)}
2) {(-5, 15), (-2, -6), (0, -10), (3, -1), (4, 6)}
Graph each data set. Which kind of model best describes the data?
CONFIDENTIAL 3
Radical ExpressionsRadical Expressions
An expression that contains a radical sign (√) is a radical expression . There are many different types of radical
expressions, but in this course, you will only study radicalexpressions that contain square roots.
The expression under a radical sign is the radicand . A radicand may contain numbers, variables, or both. It may
contain one term or more than one term.
Examples of radical expressions:
14 l2 + w2 2gd d 5√2 18 4
CONFIDENTIAL 4
Simplest Form of a Square-Root ExpressionSimplest Form of a Square-Root Expression
An expression containing square roots is in simplest form when
• the radicand has no perfect square factors other than 1.
• the radicand has no fractions.
• there are no square roots in any denominator.
CONFIDENTIAL 5
Remember that positive numbers have two square roots, one positive and one negative. However, √1 indicates a non-negative square root. When you simplify, be sure that your answer is not negative. To simplify √x2 , you
should write √x2 = |1| , because you do not know whether x is positive or negative.
Below are some simplified square-root expressions:√x2 = |x|
√x3 = x√x
√x4 = x2
√x5 = x2√x
√x6 = |x3|
CONFIDENTIAL 6
Simplifying Square-Root ExpressionsSimplifying Square-Root Expressions
Simplify each expression.
A) 2 = 1 = 1 72 36 6
B) 32 + 42 = 9 + 16 = 25 = 5
C) x2 + 8x + 16 = (x + 4)2 = |x + 4|
CONFIDENTIAL 7
Now you try!
Simplify each expression.
1a) 256 4
1b) 40 + 9
1c) 52 + 122
1d) (3 - x)2
CONFIDENTIAL 8
Product Property of Square RootsProduct Property of Square Roots
For any nonnegative real numbers a and b, the square root of ab is equal to the square root of
a times the square root of b.
WORDS
NUMBERS
ALGEBRA
CONFIDENTIAL 9
Using the Product Property of Square RootsUsing the Product Property of Square Roots
Simplify. All variables represent nonnegative numbers.
Factor the radicand using perfect squares.
Product Property of Square Roots
Product Property of Square Roots
Simplify.
Product Property of Square Roots
Since y is nonnegative, √y2 = y.
A) 18 = 9(2) = 9 (2)
= 3 (2)
B) x4y3 = x4 (y3) = x4 y2 y
= x2y y
CONFIDENTIAL 10
Simplify. All variables represent nonnegative numbers.
Now you try!
2a) 128
2b) x3 y2
2c) 48a2b
CONFIDENTIAL 11
Quotient Property of Square RootsQuotient Property of Square Roots
For any real numbers a and b (a ≥ 0 and b > 0) , the square root of a is equal to the
b square root of a divided by the square root of b.
WORDS
NUMBERS
ALGEBRA
CONFIDENTIAL 12
Using the Quotient Property of Square RootsUsing the Quotient Property of Square Roots
Simplify. All variables represent nonnegative numbers.
Quotient Property of Square Roots.
Simplify.
A) 5 = 5 9 9 = 5 3
Quotient Property of Square Roots.
Simplify.
B) a5 = a4
81a 81 = a4
81
= a2
9
Simplify.
CONFIDENTIAL 13
Simplify. All variables represent nonnegative numbers.
Now you try!
3a) 12 27
3b) 36 x4
3c) y6
4
CONFIDENTIAL 14
Using the Product and Quotient Using the Product and Quotient Properties TogetherProperties Together
Simplify. All variables represent nonnegative numbers.
Quotient Property
Write 80 as 16 (5) .
Product Property
Simplify.
a) 80 25
= 80 25
= 16(5) 25
= 16 (5) 25
= 4 (5) 5
CONFIDENTIAL 15
Quotient Property
Write 80 as 16 (5) .
Product Property
Simplify.
b) 4x5
9
= 4x5
9
= 4(x5) 9
= 4 (x4) (x) 9
= 4x4 (5) 3
CONFIDENTIAL 16
Simplify. All variables represent nonnegative numbers.
Now you try!
4a) 20 49
4b) z5
25y2
4c) p6
q10
CONFIDENTIAL 17
Sports ApplicationSports Application
A baseball diamond is a square with sides of 90 feet. How far is a throw from third base to first base? Give the
answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot.
The distance from third base to first base is the hypotenuse of a
right triangle.Use the Pythagorean Theorem:
c2 = a2 + b2
Solve for c.
Substitute 90 for a and b.
c = a2 + b2
c = 902 + 902
CONFIDENTIAL 18
Factor 16,200 using perfect squares.
Simplify.
Use the Product Property of Square Roots.
Use a calculator and round to the nearest tenth.
c = 8100 + 8100
c = 16,200
c = 100(81)(2)
c = 100 (81) (2)
c = 10 (9) (2)
c = 90 (2)
c ≈ 127.3
Simplify.
The distance is 90√2 , or about 127.3, feet.
CONFIDENTIAL 19
5) A softball diamond is a square with sides of 60 feet. How long is a throw from third base to first base in softball? Give the answer as a radical expression in simplest form. Then estimate the length to the
nearest tenth of a foot.
Now you try!
CONFIDENTIAL 20
Assessment
1) In the expression, 3x - 6 + 7, what is the radicand ?
2) Your boat is traveling due north from a dock. Your friend’s boat left at the same time from the same dock and is headed due east.
After an hour, your friend calls and tells you that he has just stopped because of engine trouble. How far must you travel to meet your friend? Give your answer as a radical expression in simplest form. Then estimate the distance to the nearest mile.
CONFIDENTIAL 21
Simplify. All variables represent nonnegative numbers.
3) 81
4) 98 2
5) (a + 7)2
6) 180
CONFIDENTIAL 22
Graph each square-root function.
7) 17 25
8) 7 16
9) 108 49
10) 204 25
CONFIDENTIAL 23
Radical ExpressionsRadical Expressions
An expression that contains a radical sign (√) is a radical expression . There are many different types of radical
expressions, but in this course, you will only study radicalexpressions that contain square roots.
The expression under a radical sign is the radicand . A radicand may contain numbers, variables, or both. It may
contain one term or more than one term.
Examples of radical expressions:
14 l2 + w2 2gd d 5√2 18 4
Let’s review
CONFIDENTIAL 24
Simplest Form of a Square-Root ExpressionSimplest Form of a Square-Root Expression
An expression containing square roots is in simplest form when
• the radicand has no perfect square factors other than 1.
• the radicand has no fractions.
• there are no square roots in any denominator.
CONFIDENTIAL 25
Remember that positive numbers have two square roots, one positive and one negative. However, √1 indicates a non-negative square root. When you simplify, be sure that your answer is not negative. To simplify √x2 , you
should write √x2 = |1| , because you do not know whether x is positive or negative.
Below are some simplified square-root expressions:√x2 = |x|
√x3 = x√x
√x4 = x2
√x5 = x2√x
√x6 = |x3|
CONFIDENTIAL 26
Simplifying Square-Root ExpressionsSimplifying Square-Root Expressions
Simplify each expression.
A) 2 = 1 = 1 72 36 6
B) 32 + 42 = 9 + 16 = 25 = 5
C) x2 + 8x + 16 = (x + 4)2 = |x + 4|
CONFIDENTIAL 27
Product Property of Square RootsProduct Property of Square Roots
For any nonnegative real numbers a and b, the square root of ab is equal to the square root of
a times the square root of b.
WORDS
NUMBERS
ALGEBRA
CONFIDENTIAL 28
Using the Product Property of Square RootsUsing the Product Property of Square Roots
Simplify. All variables represent nonnegative numbers.
Factor the radicand using perfect squares.
Product Property of Square Roots
Product Property of Square Roots
Simplify.
Product Property of Square Roots
Since y is nonnegative, √y2 = y.
A) 18 = 9(2) = 9 (2)
= 3 (2)
B) x4y3 = x4 (y3) = x4 y2 y
= x2y y
CONFIDENTIAL 29
Quotient Property of Square RootsQuotient Property of Square Roots
For any real numbers a and b (a ≥ 0 and b > 0) , the square root of a is equal to the
b square root of a divided by the square root of b.
WORDS
NUMBERS
ALGEBRA
CONFIDENTIAL 30
Using the Quotient Property of Square RootsUsing the Quotient Property of Square Roots
Simplify. All variables represent nonnegative numbers.
Quotient Property of Square Roots.
Simplify.
A) 5 = 5 9 9 = 5 3
Quotient Property of Square Roots.
Simplify.
B) a5 = a4
81a 81 = a4
81
= a2
9
Simplify.
CONFIDENTIAL 31
Using the Product and Quotient Using the Product and Quotient Properties TogetherProperties Together
Simplify. All variables represent nonnegative numbers.
Quotient Property
Write 80 as 16 (5) .
Product Property
Simplify.
a) 80 25
= 80 25
= 16(5) 25
= 16 (5) 25
= 4 (5) 5
CONFIDENTIAL 32
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