1. These two expressions are equivalent
6.2 rational exponents
Recall: Square Roots e.g. 9
√9 = 3 because 3 x 3 = 9, or 32 = 9, so:
√9 = 3
9 = 32
ROOT
These two expressions are equivalent
RADICAL

2. Rational exponents cont’d
√9 = 3 9 = 32 2
That’s why it’s called a SQUARE root, because of the “2”
There’s actually an invisible “2” where the radical is…we just got too lazy to write it.

3. Rational exponents cont’d
NAME
NUMBER
RADICAL
ROOT
Square 2
𝟐 𝒃 =𝒂
𝒂×𝒂=𝒃
Cube 3
𝟑 𝒃 =𝒂
𝒂×𝒂×𝒂=𝒃
Quartic 4
𝟒 𝒃 =𝒂
𝒂×𝒂×𝒂×𝒂=𝒃
n-th n
𝒏 𝒃 =𝒂
𝒂×𝒂×𝒂(n-times) =𝒃
EX. 𝟑 𝟖 =𝒂
i.e. the cube root of 8 is a number “a” that when multiplied by itself 3 times gives 8:
a x a x a = 8
a = 2 because 2 x 2 x 2 = 8.

4. Pop quiz!
Inspect the 2 radicals below. Can only one of them be evaluated, or can both? Explain.
𝑨 3 −8 𝑩 4 −16

5. Pop quiz results 𝑨 3 −8 𝑩 4 −16 (a) x (a) x (a) = -8
𝑨 3 −8
(a) x (a) x (a) = -8
a = -2 since (-2)(-2)(-2) = -8
𝑩 4 −16
(-a) x (-a) x (-a) x (-a) = -16
Cannot be evaluated because a negative number multiplied by itself four times (even) will not return a negative number

6. Rational exponents cont’d
Radicals aren’t always nice, whole numbers:
e.g ≅2.714
e.g. 3 −52 ≅−3.733
Now we know what radicals are, what about rational exponents?
2 4
𝟏 𝟐
…is the same as… 4
RADICAL
RATIONAL EXPONENT

7. Rational exponents cont’d
RADICAL
RATIONAL EXPONENT
2 4
4 1 2 ∛8
8 1 3
5 −3125
(−3125) 1 5
256 −1 4

8. SEE FOR YOURSELF Square the following expression: 𝑠𝑞𝑢𝑎𝑟𝑒 2 1 2
𝑠𝑞𝑢𝑎𝑟𝑒
= > Power of a power:
multiply exponents
= ×2 =
= 2 1 =2

9. SEE FOR YOURSELF cont’d
Square the following expression:
𝑠𝑞𝑢𝑎𝑟𝑒 2
= 2 x 2
= 2
…same result as squaring !
Therefore, radical signs and rational exponents are interchangeable.
𝑏 = 𝑏 , for a positive base b

10. Pop quiz!
Inspect the rational exponent powers below. Can only one of them be evaluated, or can both? Write in radical form and explain.
𝑨 −𝟑𝟐 𝟏 𝟓 𝑩 −𝟑𝟔 𝟏 𝟐

11. Pop quiz results 𝑨 −𝟑𝟐 𝟏 𝟓 in radical form is:
5 −32
= -2 since (-2)(-2)(-2)(-2)(-2) = -32
𝑩 −𝟑𝟔 𝟏 𝟐 in radical form is:
2 −36
Cannot be evaluated because you cannot take the square root of a negative number

12. Rational exponents cont’d
What if the exponents aren’t in the form 1 𝑛 , but rather are a fraction like 5 7 , or −9 4 ?
To evaluate a power with a rational exponent of the form 𝑚 𝑛 :
𝑏 𝑚 𝑛 = 𝑛 𝑏 𝑚

13. 𝒎 𝒏 rational exponent (A) 𝟐𝟕 𝟐 𝟑 (B) 𝟏𝟔 𝟓 𝟒 = 27 1 3 ×2 = 16 1 4 ×5
Write in radical form then solve:
(A) 𝟐𝟕 𝟐 𝟑
= ×2 = =
= (3) 2
= 9
(B) 𝟏𝟔 𝟓 𝟒
= ×5 = =
= (2) 5
= 32

14. 6.2 rational exp homework
p. 359#1, 3b, 4abd, 5abd, 7cdef, 8, 9, 11, 13abc