1. The “Nth” Root 𝑛 𝑎

2. 1. √121 2. 5–2 Lesson 6.1, For use with pages 414-419
Evaluate without using a calculator.
1. √121
ANSWER 11
2. 5–2
ANSWER 1 25

3. 3. Solve (x + 7)2 = 24. Write answers with radicals in simplest form.
Lesson 6.1, For use with pages
Evaluate without using a calculator.
3. Solve (x + 7)2 = 24. Write answers with radicals in
simplest form.
ANSWER
–7 ± 2√6
4. A ball has a surface area s of 1253 square inches. use the formula r = √s to find the radius r of the ball. 1
3.54
ANSWER
about 10 in.

4. EXAMPLE 1
Find nth roots
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

5. EXAMPLE 1
Find nth roots
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

6. Rational Exponent Form Radical Form
EXAMPLE 2
Evaluate expressions with rational exponents
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. Expression Keystrokes Display EXAMPLE 3
Approximate roots with a calculator
Expression
Keystrokes
Display
a /5
b /8 7
c. ( = 73/4 

8. GUIDED PRACTICE
for Examples 1, 2 and 3
Find the indicated real nth root(s) of a.
1. n = 4, a = 625
3. n = 3, a = –64.
SOLUTION ±5
SOLUTION –4
2. n = 6, a = 64
4. n = 5, a = 243
SOLUTION ±2
SOLUTION 3

9. GUIDED PRACTICE
for Examples 1, 2 and 3
Evaluate expressions without using a calculator.
5. 45/2 /4 27
SOLUTION 32
SOLUTION
–1/2
8. 17/8 1 3
SOLUTION
SOLUTION 1

10. Expression GUIDED PRACTICE for Examples 1, 2 and 3
Evaluate the expression using a calculator. Round the result to two decimal places when appropriate.
Expression /5
SOLUTION
1.74 /3 –
SOLUTION
0.06
(4√ 16)5
SOLUTION 32
(3√ –30)2
SOLUTION
9.65

11. Take fifth root of each side.
EXAMPLE 4
Solve equations using nth roots
Solve the equation.
a. 4x5
= 128 x5 32 =
Divide each side by 4. x =  32 5
Take fifth root of each side. x 2 =
Simplify.

12. Take fourth roots of each side.
EXAMPLE 4
Solve equations using nth roots
b. (x – 3)4
= 21
x – 3 4 + –  21 =
Take fourth roots of each side. x  4 + – 21
+ 3 =
Add 3 to each side. x 4  21
+ 3 = or –
Write solutions separately. x
5.14 or
0.86
Use a calculator.

13. EXAMPLE 5
Use nth roots in problem solving
Biology
A study determined that the weight w (in grams) of coral cod near Palawan Island, Philippines, can be approximated using the model
w = ℓ 3
where l is the coral cod’s length (in centimeters). Estimate the length of a coral cod that weighs 200 grams.

14. Take cube root of each side.
EXAMPLE 5
Use nth roots in problem solving
SOLUTION w
l 3 =
Write model for weight.
200
l 3 =
Substitute 200 for w.
11,976
l 3
Divide each side by
11,976 3  l
Take cube root of each side.
22.9 l
Use a calculator.
ANSWER
A coral cod that weighs 200 grams is about 23 centimeters long.

15. GUIDED PRACTICE
for Examples 4 and 5
Solve the equation. Round the result to two decimal places when appropriate.
16. 1 4
x3 = 2
x3 = 64
SOLUTION x
= 4
SOLUTION x
= 2
14. 1 2
x5 = 512
17.
( x – 2 )3 = –14
SOLUTION x
= –0.41
SOLUTION x
= 4
15.
3x2 = 108
18.
( x + 5 )4 = 16
SOLUTION
SOLUTION x –
= + 6 x
= – 3 or
= –7

16. GUIDED PRACTICE
for Examples 4 and 5
WHAT IF? Use the information from Example 5 to estimate the length of a coral cod that has the given weight.
19.
a grams
SOLUTION
25 cm
b grams
SOLUTION
27 cm
c grams
SOLUTION
30 cm