Numbers and operations · Ages 13–14

Working with integral indices worksheet

Part of: Laws of integral indices

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Numbers and operations · Ages 13–14

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Fill in the index: x⁻⁹ ÷ x³ = x^□
Answer:
2.Fill in the index: (x⁻⁶)³ = x^□
Answer:
3.Fill in the index: x⁻⁷ × x⁻⁷ = x^□
Answer:
4.Fill in the index: x⁻⁶ ÷ x⁵ = x^□
Answer:
5.Simplify, with positive indices: x⁴y⁻⁶ × x⁶y³ = x^□y^□
Fill in each blank
Answer:
6.With a positive index: x⁵ × x⁻⁸ = 1x^□
Answer:
7.Fill in the indices: (x⁻⁵y⁻³)⁻² ÷ x³y⁻³ = x^□y^□
Fill in each blank
Answer:
8.With a positive index: (x³5)⁻² = x^□
Fill in each blank
Answer:
9.With a positive index: (x²2)⁻² = x^□
Fill in each blank
Answer:

13.Simplify (x⁻²)³ × x⁴. First (x⁻²)³ = x^□; then times x⁴ gives x^□.
Fill in each blank
Working:
Answer:
14.1 micrometre is 10⁻⁶ m and 1 nanometre is 10⁻⁹ m. How many nanometres is 864 micrometres?
Working:
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15.Someone simplifies x⁶ ÷ x⁻². Which step is wrong? What is the index of x in the right answer?
1. (1)Dividing subtracts the indices
2. (2)6 − 2 = 4
3. (3)So the answer is x⁴
Find the wrong step, then write the right answer
Working:
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16.A computer takes 10⁻⁸ s for each operation. 10³ operations take 10^□ s.
Working:
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17.A drop of water is about 10⁻⁷ litres, and a pool holds about 10⁵ litres. About how many drops is the pool? It is 10ⁿ drops: what is n?
Working:
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18.Simplify (x³)⁻² × x³ ÷ x⁻⁹. Put the steps in order.
- A. x⁻⁶ × x³ ÷ x⁻⁹ = x⁻³ ÷ x⁻⁹
- B. (x³)⁻² × x³ ÷ x⁻⁹ = x⁻⁶ × x³ ÷ x⁻⁹
- C. x⁻³ ÷ x⁻⁹ = x⁶
Write the letters in the right order
Working:
Answer:

Show the answers and working

Working with integral indices worksheet: Answers and working

1. −12
Subtract the indices as directed numbers: (−9) − (3) = −12.
2. −18
Multiply the indices as directed numbers: (−6) × (3) = −18.
3. −14
Add the indices as directed numbers: (−7) + (−7) = −14.
4. −11
Subtract the indices as directed numbers: (−6) − (5) = −11.
5. 10, 3
x's index: 4 + (6) = 10; y's: (−6) + (3) = −3, and y⁻³ = 1y³.
6. 3
5 + (−8) = −3, and x⁻³ = 1x³.
7. 7, 9
x: (−5) × (−2) − (3) = 7; y: (−3) × (−2) − (−3) = 9.
8. 25, 6
Turn the fraction over first: (x³5)⁻² = (5x³)² = 25x⁶.
9. 4, 4
Turn the fraction over first: (x²2)⁻² = (2x²)² = 4x⁴.

10. False
4 − (−2) = 4 + 2 = 6: taking away a negative adds.
11. C. 1x¹²
(−3) × 3 − 3 = −12, and x⁻¹² = 1x¹².
12. False
A power of a product works for negative indices too: every factor is raised to −3, (xy)⁻³ = x⁻³y⁻³.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. −6, −2
A power of a power multiplies the indices: (−2) × (3) = −6; then multiplying adds them: −6 + (4) = −2.
14. 864000 nm
1 micrometre = 10⁻⁶ ÷ 10⁻⁹ = 10^(−6 − (−9)) = 10³ nm, so 864 micrometres = 864,000 nm.
15. 8
What is taken away is −2: 6 − (−2) = 6 + 2 = 8.
16. −5
Time = 10³ × 10⁻⁸ = 10^(3 + (−8)) = 10⁻⁵ s.
17. 12
Drops = 10⁵ ÷ 10⁻⁷ = 10^(5 − (−7)) = 10¹².
18. B → A → C
The power first: (3) × (−2) = −6; then left to right: −6 + (3) = −3 and −3 − (−9) = 6.

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