Download Section 8.2 Rational Exponents

11/13/2012
Section 8.2 Rational Exponents
Use exponential notation for nth roots.
a1/n
If
n
1/ n
n
a is a real number, then a = a .
Notice that the denominator of the rational exponent is the index of the radical.
Evaluate each exponential
Evaluate each exponential.
CLASSROOM
EXAMPLE 1
Evaluating Exponentials of the Form a1/n
Evaluate each exponential.
Solution:
321/ 5
= 5 32
=2
641/ 2
= 2 64
= 64
−811/ 4
= − 4 81
= −3
(−81)1/ 4
= 4 −81
Is not a real number because the radicand,
–81,, is negative
g
and the index,, 4,, is even.
(−64)1/ 3
= 3 −64
= −4
1/ 3
⎛ 1 ⎞
⎜ ⎟
⎝ 27 ⎠
=
3
1
27
=
=8
1
3
Slide 8.2- 2
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Evaluate each exponential.
1211/ 2 = 2 121 = 121 = 11
1/ 3
⎛ 8 ⎞
⎜ ⎟
⎝ 27 ⎠
(−36))1/ 2
(−32)1/ 5
If m and n are positive integers with m/n in lowest terms, then am/n= (a1/n)m, provided that a1/n is a real number. If a1/n
is not a real number, then am/n is not a real number.
643/ 2
2162 / 3
−323/ 5
⎛ 1 ⎞
If am/n is a real number, then a‐m/n = . ⎜
⎟ (a ≠ 0)
⎝a ⎠
m/n
27 −4 / 3
81−3/ 2
⎛ 64 ⎞
⎜
⎟
⎝ 125 ⎠
−2 / 3
.
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If all indicated roots are real numbers, then
a m / n= n a m =
( a)
n
m
In words, raise a to the mth
In
words raise a to the mth power and then take the nth power and then take the nth
root, or take the nth root of a and then raise to the mth
power. Write with radicals. Assume that all variables represent positive real numbers
positive real numbers. 31/ 2
72/3
(3p))3/4 + (4x)
(3
( )1/3
(5y)‐3/5
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(r + 2z)3/2
(5x2 + 3z3)‐5/6
Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers.
p
p
510
4
68
r 50
y ⋅ 5 y2
4
w3
6
w
4
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Simplify each expression. Write all answers with positive exponents. Assume that all variables represent positive real numbers.
64/3∙ 6
62/3
1257 / 3
1255/ 3
r‐8/9 ∙ r17/9
X2/5 ∙ x‐1/3
z 3/ 4
z 5/ 4 ⋅ z −2
(a b )
(a b )
2 5
−3 2
−1/ 4
1/ 6
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⎛ 2−2 w−3/ 4 x −5 / 8 ⎞
⎜
⎟
3/ 4 −1/ 2
⎝ w x
⎠
(
r 3/ 5 r1/ 2 + r 3/ 4
(
−3
)
4m5 / 3 m −2 / 3 − 4m −5 / 3
)
Write radicals as exponentials, and then apply the rules for rational exponents. Give answers in exponential form.
5
x3 ⋅ 4 x
x5
x8
6