Numbers and operations · Ages 13–14
Laws of integral indices worksheet
Covers: Multiplying and dividing powers, Powers of powers, products and quotients, Zero and negative indices, Working with integral indices, Writing powers with the same base, Scientific notation, Binary numbers, Hexadecimal numbers
In the print window, choose “Save as PDF” to download it. The answers print on the last page.
Laws of integral indices worksheet
Numbers and operations · Ages 13–14
Numsprite
numsprite.com
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.Simplify: 6x6y3 × 6x6y2 = x□y□
Fill in each blank
Answer:
2.Fill in the index: 511 × 54 ÷ 52 = 5□
Answer:
3.Simplify: (x3y4)2 = x□y□
Fill in each blank
Answer:
4.320 × 11 =
Answer:
5.With a positive index: x2 × x−11 = 1x□
Answer:
6.With a positive index: (x42)−2 = x□
Fill in each blank
Answer:
7.As a power of 2: 162 ÷ 84 = 2□
Answer:
8.As a decimal: 1.1 × 10−2 =
Answer:
9.As a decimal number: 110000₂ =
Answer:
10.As a decimal number: E7₁₆ − 33₁₆ =
Answer:
Part B: Choose, or true or false (2 marks each)
11.(3x2)2 = 3x4
True or false?
Answer:
12.(−6)−2 = ?
- A. −136
- B. 136
- C. 36
- D. −36
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.A rectangle is 3x3 cm long and 3x5 cm wide. Its area is x□ cm².
Fill in each blank
Working:
Answer:
14.Someone simplifies (3x4)2 and gets 3x8. Which step is wrong? What is the number in front in the right answer?
- (1)Multiply the indices: 4 × 2 = 8
- (2)Copy the number 3
- (3)So the answer is 3x8
Find the wrong step, then write the right answer
Working:
Answer:
15.Simplify (x−4)3 × x4 ÷ x8. Put the steps in order.
- A. x−12 × x4 ÷ x8 = x−8 ÷ x8
- B. x−8 ÷ x8 = x−16
- C. (x−4)3 × x4 ÷ x8 = x−12 × x4 ÷ x8
Write the letters in the right order
Working:
Answer:
16.Chi-keung says 96 = 275. Chi-keung is right.
True or false?
Working:
Answer:
17.Change 15 into binary by dividing by 2 again and again. Put the steps in order.
- A. 1 ÷ 2 = 0 r 1
- B. 3 ÷ 2 = 1 r 1
- C. 7 ÷ 2 = 3 r 1
- D. 15 ÷ 2 = 7 r 1
Write the letters in the right order
Working:
Answer:
18.The colour code #AF2F48 has red, green and blue, as decimal numbers, of , and .
Fill in each blank
Working:
Answer:
Show the answers and working
Laws of integral indices worksheet: Answers and working
Part A: Working out and filling in
1. 36, 12, 5
The numbers: 6 × 6 = 36; x's index 6 + 6 = 12; y's index 3 + 2 = 5. Each letter on its own.
2. 13
Add the indices multiplied and subtract the one divided: 11 + 4 − 2 = 13.
3. 6, 8
A power of a quotient raises the top and the bottom: 3 × 2 = 6 and 4 × 2 = 8.
4. 11
Any number but 0 to the power 0 is 1: 320 = 1, and 1 × 11 = 11.
5. 9
2 + (−11) = −9, and x−9 = 1x9.
6. 4, 8
Turn the fraction over first: (x42)−2 = (2x4)2 = 4x8.
7. −4
2^(4 × 2) ÷ 2^(3 × 4): subtract the indices, −4.
8. 0.011
Times 10−2 is dividing by 102: the point moves 2 places left, 0.011.
9. 48
From the right the place values are 1, 2, 4, 8, 16, 32: add those under the 1s, 48.
10. 180
As decimals first: 231 − 51 = 180.
Part B: Choose, or true or false
11. False
The 3 in the bracket is raised too: 32 = 9, so (3x2)2 = 9x4.
12. B. 136
(−6)−2 = 1(−6)2 = 136.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 9, 8
Area = length × width = 3x3 × 3x5: 3 × 3 = 9 and 3 + 5 = 8, so 9x8 cm².
14. 9
A power of a product raises every factor: 32 = 9, so 9x8.
15. C → A → B
The power first: (−4) × (3) = −12; then left to right: −12 + (4) = −8 and −8 − (8) = −16.
16. False
Both as powers of 3: 96 = 3^(2 × 6) and 275 = 3^(3 × 5); they are equal only when the indices are.
17. D → C → B → A
Divide each quotient by 2 again until it is 0; read the remainders from the bottom up: 15 = 1111₂.
18. 175, 47, 72
In pairs: AF₁₆ = 175, 2F₁₆ = 47, 48₁₆ = 72.
Numsprite · numsprite.com/en/worksheets/laws-of-integral-indices
Practise it in the gameLaws of integral indices · Unlimited questions, no download