Numbers and operations · Ages 13–14
Powers of powers, products and quotients worksheet
Part of: Laws of integral indices
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Powers of powers, products and quotients worksheet
Numbers and operations · Ages 13–14
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.Fill in the index: (x3)2 = x□
Answer:
2.Fill in the index: (27)2 = 2□
Answer:
3.Fill in the index: (x9)6 = x□
Answer:
4.Fill in the index: (58)2 = 5□
Answer:
5.Simplify: (4x4)2 = x□
Fill in each blank
Answer:
6.Simplify: (x5y4)4 = x□y□
Fill in each blank
Answer:
7.Fill in the index: (7x7)2 × x5 = 49x□
Answer:
8.Simplify: (x4y6)4 = x□y□
Fill in each blank
Answer:
Part B: Choose, or true or false (2 marks each)
9.(x9)6 = x54
True or false?
Answer:
10.(2x5)3 = ?
- A. 2x15
- B. 8x8
- C. 8x15
- D. 6x15
Choose the right answer
Answer:
11.(3x6)2 = ?
- A. 9x12
- B. 9x8
- C. 3x12
- D. 6x12
Choose the right answer
Answer:
12.(2x3)3 = 8x9
True or false?
Answer:
Part C: Word problems (4 marks each)
13.A cube has edges of x7 cm. Ka-yee says its volume is x21 cm³. Ka-yee is right.
True or false?
Working:
Answer:
14.A cube has edges of 3x3 cm. Its volume is x□ cm³.
Fill in each blank
Working:
Answer:
15.A square has sides of 7x3 cm. Its area is x□ cm².
Fill in each blank
Working:
Answer:
16.Someone simplifies (5x6)2 and gets 5x12. Which step is wrong? What is the number in front in the right answer?
- (1)Multiply the indices: 6 × 2 = 12
- (2)Copy the number 5
- (3)So the answer is 5x12
Find the wrong step, then write the right answer
Working:
Answer:
17.Simplify (3x6)2 × x5. Put the steps in order.
- A. 9x12 × x5 = 9x17
- B. (3x6)2 × x5 = 32 × (x6)2 × x5
- C. 32 × (x6)2 × x5 = 9x12 × x5
Write the letters in the right order
Working:
Answer:
18.A square has an area of 64x12 cm². How long is each side?
- A. 8x6 cm
- B. 64x6 cm
- C. 32x6 cm
- D. 8x10 cm
Choose the right answer
Working:
Answer:
Show the answers and working
Powers of powers, products and quotients worksheet: Answers and working
Part A: Working out and filling in
1. 6
2 lots of x3 multiplied: multiply the indices, 3 × 2 = 6.
2. 14
A power of a power multiplies the indices: 7 × 2 = 14.
3. 54
6 lots of x9 multiplied: multiply the indices, 9 × 6 = 54.
4. 16
A power of a power multiplies the indices: 8 × 2 = 16.
5. 16, 8
Every factor in the bracket is raised: 42 = 16 and (x4)2 = x8.
6. 20, 16
Both x and y are raised: 5 × 4 = 20 and 4 × 4 = 16.
7. 19
The power first: (7x7)2 = 49x^(2 × 7); then multiply, adding the indices: 2 × 7 + 5 = 19.
8. 16, 24
A power of a quotient raises the top and the bottom: 4 × 4 = 16 and 6 × 4 = 24.
Part B: Choose, or true or false
9. True
(x9)6 is 6 lots of x9 multiplied, so the index is 9 × 6 = 54, not a sum.
10. C. 8x15
Raise every factor: 23 = 8 and (x5)3 = x15, so 8x15.
11. A. 9x12
Raise every factor: 32 = 9 and (x6)2 = x12, so 9x12.
12. True
The 2 in the bracket is raised too: 23 = 8, so (2x3)3 = 8x9.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. True
Volume = (x7)3 = x^(7 × 3) = x21 cm³: a power of a power multiplies the indices.
14. 27, 9
Volume = (3x3)3 = 33 × (x3)3 = 27x9 cm³.
15. 49, 6
Area = (7x3)2 = 72 × (x3)2 = 49x6 cm².
16. 25
A power of a product raises every factor: 52 = 25, so 25x12.
17. B → C → A
Raise every factor in the bracket first, then multiply powers of one base: 2 × 6 + 5 = 17.
18. A. 8x6 cm
(8x6)2 = 64x12, so each side is 8x6 cm.
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