10.1 Radical Expressions and Graphs
• is the positive square root of a, andis the negative square root of a because
• If a is a positive number that is not a perfect square then the square root of a is irrational.
• If a is a negative number then square root of a is not a real number.
• For any real number a:
aaaa 22
and
a a
aa 2
10.1 Radical Expressions and Graphs
• The nth root of a:
is the nth root of a. It is a number whose nth power equals a, so:
• n is the index or order of the radical
• Example:
aann
n a
322 because 2 32 55
10.1 Radical Expressions and Graphs
• The nth root of nth powers:– If n is even, then
– If n is odd, then
• The nth root of a negative number:– If n is even, then the nth root is not a real number– If n is odd, then the nth root is negative
aan n
n na
aan n
10.1 - Graph of a Square Root Function
(0, 0)
xxf )(
10.2 Rational Exponents
• Definition:
• All exponent rules apply to rational exponents.
mnnm
nm
mn
m
nnm
nn
aaa
aaa
aa
11
1
1
10.2 Rational Exponents
• Tempting but incorrect simplifications:
nm
nm
nm
n
aaaaa
aa
n
m
n
11
10.2 Rational Exponents
• Examples:
51
251
2512525
2525
2555555
21
21
43
41
43
41
36
34
32
34
32 2
10.3 Simplifying Radical Expressions
• Review: Expressions vs. Equations:– Expressions
1. No equal sign2. Simplify (don’t solve)3. Cancel factors of the entire top and bottom of a fraction
– Equations1. Equal sign2. Solve (don’t simplify)3. Get variable by itself on one side of the equation by
multiplying/adding the same thing on both sides
10.3 Simplifying Radical Expressions
• Product rule for radicals:
• Quotient rule for radicals:
nnn abba
nn
n
ba
ba
10.3 Simplifying Radical Expressions
• Example:
• Example:
416348
348
333
3
81 81 27 333
10.3 Simplifying Radical Expressions
• Simplified Form of a Radical:1. All radicals that can be reduced are reduced:
2. There are no fractions under the radical.3. There are no radicals in the denominator4. Exponents under the radical have no common
factor with the index of the radical
aaaa 21
42
4 2
33 4 and 39 aaa
10.3 Simplifying Radical Expressions
• Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2
• Pythagorean triples (integer triples that satisfy the Pythagorean theorem): {3, 4, 5}, {5, 12, 13}, {8, 15, 17}
a
b
c
90
10.3 Simplifying Radical Expressions
• Distance Formula: The distance between 2 points (x1, y1) and (x2,y2) is given by the formula (from the Pythagorean theorem):
212
212 yyxxd
10.4 Adding and Subtracting Radical Expressions
• We can add or subtract radicals using the distributive property.
• Example:
373)25(3235
10.4 Adding and Subtracting Radical Expressions
• Like Radicals (similar to “like terms”) are terms that have multiples of the same root of the same number. Only like radicals can be combined.
383335
393539352735
combined becannot 323
combined becannot 5233
10.4 Adding and Subtracting Radical Expressions
• Tempting but incorrect simplifications:
2222 yxyx
yxyx
10.5 Multiplying and Dividing Radical Expressions
• Use FOIL to multiply binomials involving radical expressions
• Example: )63)(25(
6637
326635
1266535
62326535
10.5 Multiplying and Dividing Radical Expressions
• Examples of Rationalizing the Denominator:
3 3 3 3 3 333 3 3
5 5 5 2 5 2 102 22 2 2 2 2
10.5 Multiplying and Dividing Radical Expressions
• Using special product rule with radicals:
2 2
223 1 3 1 3 1 3 1 2
a b a b a b
10.5 Multiplying and Dividing Radical Expressions
• Using special product rule for simplifying a radical expression:
22
2 3 12 2 3 13 1 3 1 3 1 3 1
2 3 1 2 3 13 1
3 1 2
10.6 Solving Equations with Radicals
• Squaring property of equality: If both sides of an equation are squared, the original solution(s) of the equation still work – plus you may add some new solutions.
• Example:
5)- and (5 solutions twohas 25
(5)solution one has 52
x
x
10.6 Solving Equations with Radicals
• Solving an equation with radicals:1. Isolate the radical (or at least one of the radicals if
there are more than one).2. Square both sides3. Combine like terms4. Repeat steps 1-3 until no radicals are remaining5. Solve the equation6. Check all solutions with the original equation (some
may not work)
10.6 Solving Equations with Radicals
• Example:Add 1 to both sides:
Square both sides:
Subtract 3x + 7:
So x = -2 and x = 3, but only x = 3 makes the original equation equal.
0)2)(3(06 2
xxxx
173 xx731 xx
73122 xxx
10.7 Complex Numbers
• Definition:
• Complex Number: a number of the form a + bi where a and b are real numbers
• Adding/subtracting: add (or subtract) the real parts and the imaginary parts
• Multiplying: use FOIL
1 and 1 2 ii
10.7 Complex Numbers
• Examples:
iiiiiiii
iiiiiiii
746342)2(3)1(3)2(2)1(2)21)(32(
33)25()14()21()54(51)23()12()21()32(
2
10.7 Complex Numbers
• Complex Conjugate of a + bi: a – bimultiplying by the conjugate:
• The conjugate can be used to do division(similar to rationalizing the denominator)
13)9(494
)3(2)32)(32(2
22
i
iii
10.7 Complex Numbers
• Dividing by a complex number:
iiii
iiiii
ii
ii
132
1323
13223
)9(4)1(1528
)3(21510128
3232
3254
3254
22
2
