1. 04 Evaluate nth Roots and Use Rational Exponents
pg. P14
What is a quick way to tell what kind of real roots you have?
How do you write a radical in exponent form?
What buttons do you use on a calculator to approximate a radical?
What is the difference between evaluating and solving?

2. Real nth Roots
Let n be an integer greater than 1 and a be a real number.
If n is odd, then a has one real nth root.
If n is even and a > 0, then a has two real nth roots.
If n is even and a = 0, then a has one nth root.
If n is even and a < o, then a has no real nth roots.

3. Find the indicated real nth root(s) of a.
a. n = 3, a = –216
b. n = 4, a = 81
SOLUTION
a. Because n = 3 is odd and a = –216 < 0, –216 has one real cube root. Because (–6)3 = –216, you can write = 3√ –216 = –6 or (–216)1/3 = –6.
b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 34 = 81 and (–3)4 = 81, you can write ±4√ 81 = ±3

4. Find the indicated real nth root
n = 3, a = −125
n = 4, a = 16

5. Rational Exponents
Let a1/n be an nth root of a, and let m be a positive integer.
Powers go up, Roots go down

6. Evaluate (a) 163/2 and (b) 32–3/5.
SOLUTION
Rational Exponent Form
Radical Form
a /2
(161/2)3 = 43 = 64 =
163/2
( )3  = 16 43 = 64 = = 1
323/5 = 1
(321/5)3 1
323/5 = 1
( )3 5  32 =
b –3/5
32–3/5 = 1 23 1 8 = = 1 23 1 8 =

7. Evaluate the expression with Rational Exponents
93/2
32-2/5

8. Approximate roots with a calculator
Expression
Keystrokes
Display
a /5 1 5
b /8 3 8 7
c. ( )3 = 73/4  3 4

9. Using a calculator to approximate a root
Rewrite the problem as 53/4 and enter using ^ or yx key for the exponent.

10. Evaluate the expression using a calculator
Evaluate the expression using a calculator. Round the result to two decimal places when appropriate.
Expression
Keystrokes
Display /5
1.74 1
(64 /3 –
0.06
(4√ 16)5 32
(3√ –30)2 –
9.65

11. Solve the equation using nth roots.
2x4 = 162
x4 = 81
x4 = 34
x = ±3
(x − 2)3 = 10
x ≈ 4.15

12. 1 2
x5 = 512
SOLUTION 1 2 x5
= 512 x5
= 1024
Multiply each side by 2. x =
take 5th root of each side. x
= 4
Simplify.

13. ( x – 2 )3 = –14
SOLUTION
( x – 2 )3
= –14
( x – 2 )
= –14 x
= – x
= – x
= – 0.41
Use a calculator.

14. ( x + 5 )4 = 16
SOLUTION
( x + 5 )4
= 16
( x + 5 ) =
take 4th root of each side.
add 5 to each side. x
= – 5 x
= 2 – 5 or
= – 2 – 5
Write solutions separately. x
= – 3 or
= –7
Use a calculator.

15. Evaluating a model with roots.
When you take a number with a rational exponent and express it in an integer answer, you have evaluated.
Solving an equation using an nth root.
When you have an equation with value that has a rational exponent, you solve the equation to find the value of the variable.

16. What is a quick way to tell what kind of real roots you have?
Root is odd, 1 answer; root is even, 1 or 2 real answers.
How do you write a radical in exponent form?
Use a fraction exponent (powers go up, roots go down)
What buttons do you use on a calculator to approximate a radical?
Root buttons
What is the difference between evaluating and solving?
Evaluating simplifies; Solving finds answers x=.