Number

Indices

See powers, reciprocals, cancellation and grouped factors.

Predict, test and explain

Step 1 of 3

Predict

What is 2⁻²?

2^{3}\times2^{2}=2^{5}

2

2

2

2

2

1

a\ne0

Multiply the factor fractions, then cancel matching nonzero numerator and denominator factors. A zero exponent contributes 1.

Check the working

Quantity	Value
Result exponent	5
Exact value	$32$
Zero index	$2^0=1$

Explore and compare

Can you break the rule?

A negative integer power of a positive base is a negative number.

Always, sometimes or never? Test it in the simulation.

Hint

Choose Divide or a negative power. Look at which factors remain in the denominator.

Check the reasoning

Never. It is a reciprocal of a positive number, so it is still positive.

$2^{-3}=\frac1{2^3}=\frac18>0$

One counterexample disproves “always”. Examples alone do not prove it.

Keep the numerator fixed. Predict what happens as the divisor grows.

Predict

Try it in the simulation. Change one thing at a time.

Hint

Choose Divide. Set base 2 and first exponent 3. Increase the second exponent from 1 through 3 to 4. Use Show working.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Simplify to an exact number.

$\frac{(2^3)^2}{2^8}$

Hint

Multiply the bracket indices, then subtract the denominator's index.

Reveal answer

One quarter. A negative index gives the reciprocal of the positive power.

$2^{3\times2-8}=2^{-2}=\frac14$

Try a short practice

Use these fixed values. The diagram controls are separate.

Check your understanding

What is the exact value?

$3^{-2}$

Complete the missing step

Complete the final step.

$(2^3)^2$

$(2\times2\times2)(2\times2\times2)$
$2^{3\times2}=2^6$

Spot the first mistake

This is an attempted solution containing a mistake.

Which step contains the first mistake?

$2^3\times2^2$

$2^{3\times2}$
$2^6$
$64$

Build an explanation

Hint 1

For the same non-zero base, multiply powers by adding indices and divide powers by subtracting indices: a^m × a^n = a^(m + n), a^m ÷ a^n = a^(m − n).

Hint 2

A power of a power multiplies the indices: (a^m)^n = a^(mn).

Hint 3

For a non-zero base, a^0 = 1 and a^(−n) = 1/a^n. A negative index means a reciprocal.

Worked example

This example uses fixed values, separate from the diagram controls.

Simplify (2³)² × 2⁻¹ ÷ 2².

Use the power rule: (2³)² = 2⁶.
Add the index when multiplying and subtract it when dividing: 2⁶ × 2⁻¹ ÷ 2² = 2^(6 − 1 − 2).
The resulting index is 3, so 2³ = 8.

Answer: 8

Connect this idea

Watch out: 2³ × 2² is 2⁵, not 2⁶. Multiply indices only when raising a power to another power.