1. 7.1 nth Roots and Rational Exponents
2/19/2014

2. nth Root
Ex. 32 = 9, then 3 is the square root of 9. If b2 = a, then b is the square root of a. If b3 = a, then b is the cube root of a. If b4 = a, then b is the fourth root of a. If bn = a, then b is the nth root of a. You can write the nth root of a as 𝑛 𝑎 Where a is a real number and n is the index of the radical.

3. Number of Real Roots (use for solving)
𝑛 𝑎
Example a
Any real number
Greater than 0
Less than 0 n
Odd
Even
Number of Roots
One
Two
No Real
Solution

4. 𝑛 𝑎 Example 1 Find the indicated nth root(s) of a. a. = n 3, a 64 – b.
Find nth Root(s)
Find the indicated nth root(s) of a. a. = n 3, a 64 – b. 4, 81
SOLUTION
a. Because n is odd, 64 has one real cube root. – 3 64 –
=−4
CHECK ( )3 4 – = ) 64
Because n is even and a is greater than 0, 81 has two real fourth roots.
=3 𝑎𝑛𝑑 −3 4 81 4

5. Extra Practice
Find the indicated nth root(s) of a. 1. n = 2, a =
144
ANSWER 12 –
12, 2. = n 3, a
1000
ANSWER 10 3. = n 4, a
256
ANSWER 4 – 4,

6. Example 2 Solve the equation. a. = 2x 4 162 SOLUTION a. = 2x 4 162 = 2
Solve Equations Using nth Roots
Solve the equation. a. =
2x 4
162
SOLUTION a. =
2x 4
162
Write original equation. = 2
162
2x 4
Divide each side by 2. =
x 4 81 4 =
x 4 81
Take fourth root of each side. = x 3 – + 6

7. x = -5 −2 𝑥 3 =250 𝑥 3 =−125 Divide both sides by -2
Example 2
Solve Equations Using nth Roots b.
−2 𝑥 3 =250
𝑥 3 =− Divide both sides by -2
3 𝑥 3 = 3 − Cube root both sides
x = -5

8. When to have 1 answer instead of 2 answers when doing problems with
When the problem says SOLVE, you may have 1 or 2 answers depending if the index is odd or even.
When the problem says EVALUATE, then you only have 1 answer.

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10. Vocabulary Rational Exponents: exponents written as fractions Ex :
Radical Form:
In general:

11. Evaluate the expression.
Example 3
Evaluate Expressions with Rational Exponents
Evaluate the expression. a.
91/2 = 9 = 3 b.
161/4 = 16 4 = 2 c.
641/3 = 64 3 = 4 (
)1/4 32 – d. = 32 4 –
, no real solution

12. Evaluate the expression.
Extra Practice
Evaluate Expressions
Evaluate the expression. 4.
251/2
ANSWER 5 5.
811/2
ANSWER 9 6.
1251/3
ANSWER 5 7.
321/5
ANSWER 2

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14. (when numerator is not 1)
Rational Exponents
(when numerator is not 1)
Ex :
Radical Form:
In general:
Note: denominator is the index of the radical and
numerator is the exponent of the radical

15. Negative Rational Exponent
negative exponent still “moves” power
Ex :
Radical Form:
In general:

16. ( ) ( ) ( ) ( ) Example 4 Rewrite using rational exponents. 3 5 a. 3 5
Rewrite Expressions
Rewrite using rational exponents. ( ) 4 3 5 a. ( ) 4 3 5 =
53/4 b.
Rewrite using radicals.
72/5
72/5 = ( ) 5 2 7 c.
Rewrite using radicals.
2 2/3 – = 1
22/3 = 1 ( ) 3 2
2 2/3 – 16

17. ( ) Extra Practice Rewrite the expression using rational exponents. 8.2
ANSWER
22/5 5 2 9. 1 13 4
ANSWER
13 1/4 –

18. ( ) ( ) Extra Practice Rewrite the expression using radicals. 10.
ANSWER 15 ( ) 3 2
152/3
ANSWER 1 11 3
11.
11 1/3 –
ANSWER 1 29 ( ) 5 2
12.
29 2/5 –

19. ( ) ( ) Example 5 Evaluate the expression. a. b. 8 2/3 – 43/2 SOLUTION
Evaluate Expressions with Rational Exponents
Evaluate the expression. a. b.
8 2/3 –
43/2
SOLUTION
Use radicals to rewrite and evaluate each expression. a.
43/2 3 = ( ) 4 = 23 8 = = 1
82/3 = 1 ( ) 3 2 8 = 1 22 = 4 1 b.
8 2/3 – 19

20. Evaluate the expression.
Checkpoint
Rewrite and Evaluate Expressions with Rational Exponents
Evaluate the expression.
17.
253/2
ANSWER
125
18.
165/4
ANSWER 32
ANSWER 1 32
19.
8 5/3 –

21. Homework: WS 7.1 Odd problems only, skip #11
“I tried to catch some fog. I mist!”