1. Evaluate nth Roots and Use Rational Exponents Lesson 3.1
Honors Algebra 2
Evaluate nth Roots and Use Rational Exponents
Lesson 3.1

2. Goals Goal Rubric Evaluate nth roots Rational exponents
Evaluate expressions with rational exponents
Solve equations with nth roots
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
nth root of a
Index of a radical

4. nth Roots
You are probably familiar with finding the square root of a number. Square and square root, these two operations are inverses of each other. Similarly, there are roots that correspond to larger powers.
5 and –5 are square roots of 25 because 52 = 25 and (–5)2 = 25
2 is the cube root of 8 because 23 = 8.
2 and –2 are fourth roots of 16 because 24 = 16 and (–2)4 = 16.
a is the nth root of b if an = b.

5. nth Roots A radical sign is used to indicate a root.
The number under the radical sign is the radicand.
The index gives the degree of the root.

6. nth Roots
The nth root of a real number a can be written as the radical expression 𝑛 𝑎 , where n is the index (plural: indices) of the radical and a is the radicand. When a number has more than one root, the radical sign indicates only the principal, or positive, root.

7. nth Roots
When a radical sign shows no index, it represents a square root.
Reading Math

8. Find the indicated real nth root(s) of a.
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 or ±811/4 = ±3.

9. Find the indicated real nth root(s) of a. 1. n = 4, a = 625
Your Turn:
for Example 1,
Find the indicated real nth root(s) of a.
1. n = 4, a = 625
4√ 625 = ±5 or ±(625)1/4 = ±5
ANSWER
2. n = 6, a = 64
6√ 64 = ±2 or ±(64)1/6 = ±2
ANSWER

10. Find the indicated real nth root(s) of a. 3. n = 3, a = –64
Your Turn:
for Example 1
Find the indicated real nth root(s) of a.
3. n = 3, a = –64
3√ –64 = –4 or (–64)1/3 = –4
ANSWER
4. n = 5, a = 243
5√ 243 = 3 or (243)1/5 = 3
ANSWER

11. A rational exponent is an exponent that can be expressed as 𝑚 𝑛 , where m and n are integers and n ≠ 0. Radical expressions can be written by using rational exponents.
Rational Exponents

12. Rational Exponents Writing Math
The denominator of a rational exponent becomes the index of the radical.
Writing Math

13. Rational Exponents
Rational exponents have the same properties as integer exponent.

14. Evaluate (a) 163/2 and (b) 32–3/5.
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 16 √
( )3 = 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 =

15. Evaluate the expressions without using a calculator. 5. 45/2
Your Turn:
for Example 2
Evaluate the expressions without using a calculator.
5. 45/2 32
ANSWER
–1/2 1 3
ANSWER

16. Evaluate the expressions without using a calculator. 7. 813/4
Your Turn:
for Example 2
Evaluate the expressions without using a calculator. /4 27
ANSWER
8. 17/8 1
ANSWER

17. Approximating Roots with a Calculator
Write the root as a rational exponent.
Use a graphing calculator to approximate:
First rewrite as Then enter the following:

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

19. Your Turn:
for Example 3
Evaluate the expression using a calculator. Round the result to two decimal places when appropriate. /5
1.74
ANSWER /3 –
0.06
ANSWER
(4√ 16)5 32
ANSWER
(3√ –30)2
9.65
ANSWER

20. Solve Equations Using nth Root
To solve simple equations involving xn;
isolate the power, xn or (x – a)n term.
take the nth root of each side.

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

22. EXAMPLE 4 b. (x – 3)4 = 21 x – 3 21 4 – = x – 21 + 3 √ x 21 + 3 = or –
Solve equations using nth roots
b. (x – 3)4
= 21
x – 3 √ 21 4 + – =
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.

23. Your Turn:
for Example 4
Solve the equation. Round the result to two decimal places when appropriate.
x3 = 64 x
= 4
ANSWER
14. 1 2
x5 = 512 x
= 4
ANSWER

24. Your Turn:
for Example 4
Solve the equation. Round the result to two decimal places when appropriate.
15.
3x2 = 108 x
= + 6 –
ANSWER
16. 1 4
x3 = 2 x
= 2
ANSWER

25. Your Turn:
for Example 4
Solve the equation. Round the result to two decimal places when appropriate.
17.
( x – 2 )3 = –14 x
= – 0.41
ANSWER
18.
( x + 5 )4 = 16
ANSWER x
= – 3 or
= –7

26. 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.

27. A coral cod that weighs 200 grams is about 23 centimeters long.
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.

28. Your Turn:
for Example 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
about 25 cm
ANSWER
b grams
about 27 cm
ANSWER
c grams
about 30 cm
ANSWER