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Powers, Roots & IndicesPowers, Roots & Indices

Sample Questions & SolutionsSample Questions & Solutions

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Powers, Roots & IndicesPowers, Roots & Indices• Use repeated multiplication to evaluate the

following: (a) 9², (b) 15², (c) 2⁴, (d) 8³, (e) 10³

• Answers:Answers:

(a)(a) 9 x 9 = 819 x 9 = 81

(b)(b) 15 x 15 = 22515 x 15 = 225

(c)(c) 2 x 2 x 2 x 2 = 162 x 2 x 2 x 2 = 16

(d)(d) 8 x 8 x 8 = 5128 x 8 x 8 = 512

(e)(e) 10 x 10 x 10 = 1,00010 x 10 x 10 = 1,000

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• 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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• (a) 5² x 5⁴ = 5⁶ (2 + 4 = 6)= 5⁶ (2 + 4 = 6)

• (b) 5³ x 5 = 5⁴ (3 + 1 = 4)= 5⁴ (3 + 1 = 4)

• (c) 7 x 7 = 7² (1 + 1 = 2)= 7² (1 + 1 = 2)

• (d) 8³ x 8³ = 8⁶ (3 + 3 = 6)= 8⁶ (3 + 3 = 6)

• (e) 4² x 4⁶ = 4⁸ (2 + 6 = 8)= 4⁸ (2 + 6 = 8)

• (f) 4³ x 4² x 4⁵ = 4¹ (3 + 2 + 5 = 10)⁰= 4¹ (3 + 2 + 5 = 10)⁰

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Powers, Roots & IndicesPowers, Roots & Indices• Express the following as a single power:

(a)6⁵ 6³

(b)7⁶ 7⁵

(c)8⁴ 8²

(d)3⁵ 3

(e)4³ 4

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Powers, Roots & IndicesPowers, Roots & Indices• Answers:Answers:

(a)6⁵ = 6²6² (5 – 3 = 2) 6³

(b)7⁶ = 77 (6 – 5 = 1) 7⁵

(c)8⁴ = 8²8² (4 – 2 = 2) 8²

(d)3⁵ = 3⁴3⁴ (5 – 1 = 4) 3

(e)4³ = 4²4² ( 3 – 1 = 2) 4

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• Write the following as a single power:

(a) (4³)²

(b) (5⁴)³

(c) (6²)⁴

(d) (5¹´²)²

(e) (8²´³)³

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(a) (4³)² = 4⁶4⁶ (3 x 2 = 6)

(b) (5⁴)³ = 5¹² 5¹² (4 x 3 = 12)

(c) (6²)⁴ = 6⁸ 6⁸ (2 x 4 = 8)

(d) (5¹´²)² = 5 5 ( ½ of 2 = 1)

(e) (8²´³)³ = 8²8² (⅔ of 3 = 2)

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• Calculate the value of the following:

(a) 16¹´²

(b) 36¹´²

(c) 3 x 81¹´²

(d) 5(49)¹´²

(e) 2(25)¹´²

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(a) 16¹´² = √16 = 44

(b) 36¹´² = √36 = 66

(c) 3 x 81¹´² = 3 x √81 = 3x 9 = 2727

(d) 5(49)¹´² = 5 x √49 = 5 x 7 = 3535

(e) 2(25)¹´² = 2 x √25 = 2 x 5 = 1010

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Powers, Roots & IndicesPowers, Roots & Indices• Simplify the following:

2⁵ x 2³

2⁴

5² x 5⁴

5³

8² x 8⁵

8⁶

7² x 7⁴

7³

3⁷ x 3

3⁵

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Powers, Roots & IndicesPowers, Roots & Indices• Answers:Answers:

2⁵ x 2³ = (5 + 3)- 4 = 8 – 4 = 4 so 2⁴ = 16so 2⁴ = 16

2⁴

5² x 5⁴ = (2 + 4)- 3 = 6 – 3 = 3 so 5³ = 125so 5³ = 125

5³

8² x 8⁵ = (2 + 5)- 6 = 7 – 6 = 1 so 8¹ = 8so 8¹ = 8

8⁶

7² x 7⁴ = (2 + 4)- 3 = 6 – 3 = 3 so 7³ = 343so 7³ = 343

7³

3⁷ x 3 = (7 + 1)- 5 = 8 – 5 = 3 so 3³ = 27so 3³ = 27

3⁵