10^−4 = 1/40 True or false?

False. 10^−4 = 1/10^4, and 10^4 is 4 lots of 10 multiplied, 10000, not 10 × 4.

As a fraction: (9/5)^−1 = □ Answer in simplest form

5/9. Turn the fraction over first: (9/5)^−1 = (5/9)^1 = 5/9.

As a fraction: 8^0 + 8^−1 + 8^−2 = □ Answer in simplest form

73/64. 8^0 = 1, 8^−1 = 1/8, 8^−2 = 1/64; over 64: 64/64 + 8/64 + 1/64 = 73/64.

Laws of integral indices · Ages 13–14

Zero and negative indices

a⁰ = 1, a⁻ⁿ = 1aⁿ; a negative index isn't a negative number.

How to do it

Method

a^0 = 1, a^−n = 1/a^n; a negative index isn't a negative number

2⁻³ = 1 ÷ 2³ = 18

Another way

Follow the pattern down

2¹ = 2 → 2⁰ = 1 → 2⁻¹ = 12 → 2⁻² = 14 → 2⁻³ = 18

In full

Look at the pattern: 2³ = 8, 2² = 4, 2¹ = 2. Each time the index goes down by 1, the value is divided by 2. Carry on: 2⁰ = 1, 2⁻¹ = 12, 2⁻² = 14, 2⁻³ = 18. So for any number a that isn't 0: a⁰ = 1 and a⁻ⁿ = 1aⁿ. Dividing powers says the same: a³ ÷ a³ = a⁰, and a number divided by itself is 1; a² ÷ a⁵ = a⁻³, and cancelling gives 1a³. A negative index doesn't make a negative number: 3⁻² = 19, which is positive. A fraction to a negative index: turn the fraction over first, (23)⁻² = (32)² = 94. Powers of 10 with negative indices are decimals: 10⁻³ = 11000 = 0.001.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
10⁻⁴ = 140
True or false?
Show the answer and working
Answer: False
10⁻⁴ = 110⁴, and 10⁴ is 4 lots of 10 multiplied, 10000, not 10 × 4.
Example 2
As a fraction: (95)⁻¹ =
Answer in simplest form
Show the answer and working
Answer: 59
Turn the fraction over first: (95)⁻¹ = (59)¹ = 59.
Example 3
As a fraction: 8⁰ + 8⁻¹ + 8⁻² =
Answer in simplest form
Show the answer and working
Answer: 7364
8⁰ = 1, 8⁻¹ = 18, 8⁻² = 164; over 64: 6464 + 864 + 164 = 7364.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.3⁻³ = 127
True or false?
Answer:
2.2⁻⁴ = ?
- A. 116
- B. 18
- C. −16
- D. −8
Choose the right answer
Answer:
3.18⁰ × 6 =
Answer:
4.7⁻⁴ = 1 ÷ 7^□ = 1
Fill in each blank
Answer:
5.As a fraction: 9⁰ + 9⁻¹ + 9⁻² =
Answer in simplest form
Answer:

All the answers

1. True
2. A. 116
3. 6
4. 4, 2401
5. 9181

Free worksheet: Zero and negative indicesWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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