1. Powers, Roots & Indices
Sample Questions & Solutions

2. Powers, Roots & Indices
Use repeated multiplication to evaluate the following: (a) 9², (b) 15², (c) 2⁴, (d) 8³, (e) 10³
Answers:
(a) 9 x 9 = 81
(b) 15 x 15 = 225
(c) 2 x 2 x 2 x 2 = 16
(d) 8 x 8 x 8 = 512
(e) 10 x 10 x 10 = 1,000

3. Powers, Roots & Indices Write the following as a single power:
(a) 5² x 5⁴
(b) 5³ x 5
(c) 7 x 7
(d) 8³ x 8³
(e) 4² x 4⁶
(f) 4³ x 4² x 4⁵

4. Powers, Roots & Indices Answers: (a) 5² x 5⁴ = 5⁶ (2 + 4 = 6)
(b) 5³ x 5 = 5⁴ (3 + 1 = 4)
(c) 7 x 7 = 7² (1 + 1 = 2)
(d) 8³ x 8³ = 8⁶ (3 + 3 = 6)
(e) 4² x 4⁶ = 4⁸ (2 + 6 = 8)
(f) 4³ x 4² x 4⁵ = 4¹⁰ ( = 10)

5. Powers, Roots & Indices Express the following as a single power: 6⁵ 6³
6⁵ ³
7⁶ ⁵
8⁴ ² 3⁵ 4³

6. Powers, Roots & Indices Answers: 6⁵ = 6² (5 – 3 = 2) 6³
6⁵ = 6² (5 – 3 = 2) ³
7⁶ = 7 (6 – 5 = 1) ⁵
8⁴ = 8² (4 – 2 = 2) ²
3⁵ = 3⁴ (5 – 1 = 4)
4³ = 4² ( 3 – 1 = 2)

7. Powers, Roots & Indices Write the following as a single power: (4³)²
(5⁴)³
(6²)⁴
(5¹´²)²
(8²´³)³

8. Powers, Roots & Indices Answers: (4³)² = 4⁶ (3 x 2 = 6)
(5¹´²)² = 5 ( ½ of 2 = 1)
(8²´³)³ = 8² (⅔ of 3 = 2)

9. Powers, Roots & Indices Calculate the value of the following: 16¹´²
36¹´²
3 x 81¹´²
5(49)¹´²
2(25)¹´²

10. Powers, Roots & Indices Answers: 16¹´² = √16 = 4 36¹´² = √36 = 6
3 x 81¹´² = 3 x √81 = 3x 9 = 27
5(49)¹´² = 5 x √49 = 5 x 7 = 35
2(25)¹´² = 2 x √25 = 2 x 5 = 10

11. Powers, Roots & Indices Simplify the following: 2⁵ x 2³ 2⁴ 5² x 5⁴ 5³8⁶
7² x 7⁴ 7³
3⁷ x 3 3⁵

12. Powers, Roots & Indices Answers:
2⁵ x 2³ = (5 + 3)- 4 = 8 – 4 = 4 so 2⁴ = 16 2⁴
5² x 5⁴ = (2 + 4)- 3 = 6 – 3 = 3 so 5³ = 125 5³
8² x 8⁵ = (2 + 5)- 6 = 7 – 6 = 1 so 8¹ = 8 8⁶
7² x 7⁴ = (2 + 4)- 3 = 6 – 3 = 3 so 7³ = 343 7³
3⁷ x 3 = (7 + 1)- 5 = 8 – 5 = 3 so 3³ = 27 3⁵