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Powers, Roots & Indices Sample Questions & Solutions

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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

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**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

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**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)

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**Powers, Roots & Indices Express the following as a single power: 6⁵ 6³**

6⁵ ³ 7⁶ ⁵ 8⁴ ² 3⁵ 4³

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**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)

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**Powers, Roots & Indices Write the following as a single power: (4³)²**

(5⁴)³ (6²)⁴ (5¹´²)² (8²´³)³

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**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)

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**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

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**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

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**Powers, Roots & Indices Simplify the following: 2⁵ x 2³ 2⁴ 5² x 5⁴ 5³**

8⁶ 7² x 7⁴ 7³ 3⁷ x 3 3⁵

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**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

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