Negative Exponents and Power of a Power

Q: What is the rule for negative exponents?

A negative exponent tells you to take the reciprocal of the base and raise it to the matching positive exponent. For example, 2⁻³ equals 1 over 2³, which equals 1/8. The minus sign flips the number into a fraction, rather than making the answer negative.

Q: Can the base of a negative exponent be negative?

Yes, the base can be negative. You still take the reciprocal and use the positive exponent, keeping the negative sign on the base. For example, (-5)⁻³ equals 1 over (-5)³, which equals 1/(-125). The method does not change when the base is negative.

Q: How do you work out a negative exponent with a fraction?

With a fraction, flip the numerator and the denominator over, then apply the positive exponent to the new fraction. For example, (2/3)⁻² becomes (3/2)², which equals 9/4. Swapping the fraction over removes the negative sign, so you can then square it as normal.

Q: Why do you multiply the powers in the power of a power rule?

Raising 2³ to the power of 5 means five sets of 2³ multiplied together. Each 2³ is three twos, so five sets give fifteen twos in total. That is why you multiply the exponents, giving 2¹⁵ rather than counting every two.

Key concept

Negative indices tell you to take the reciprocal, so the base flips into a fraction with a positive power. For example, 2⁻³ = 1/2³ = 1/8. A power of a power means you multiply the indices, giving (2³)⁵ = 2¹⁵.

Negative Exponents and Power of a Power - introduction visual

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Laws of indices showing a⁻ᵐ = 1/aᵐ and (aᵐ)ⁿ = aᵐⁿ for negative indices and power of a power

Negative indices rules shown with examples.

Power of a power rule with examples using positive and negative exponents.

$Power of a power rule illustrated with examples using positive, negative and fractional exponents.$ Laws of indices showing negative indices and power of a power, with examples including negative powers and simplification of expressions.

Laws of indices showing negative indices and power of a power, with examples including negative powers and simplification of expressions.

Negative Indices: Make a Reciprocal

A negative index means take the reciprocal: $a^{- m} = \frac{1}{a ^{m}}$ .
Example: $2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$ .

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What is $(2^{3})^{5}$ ?

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What is the rule for negative exponents?

A negative exponent tells you to take the reciprocal of the base and raise it to the matching positive exponent. For example, 2⁻³ equals 1 over 2³, which equals 1/8. The minus sign flips the number into a fraction, rather than making the answer negative.

Can the base of a negative exponent be negative?

Yes, the base can be negative. You still take the reciprocal and use the positive exponent, keeping the negative sign on the base. For example, (-5)⁻³ equals 1 over (-5)³, which equals 1/(-125). The method does not change when the base is negative.

How do you work out a negative exponent with a fraction?

With a fraction, flip the numerator and the denominator over, then apply the positive exponent to the new fraction. For example, (2/3)⁻² becomes (3/2)², which equals 9/4. Swapping the fraction over removes the negative sign, so you can then square it as normal.

What is the power of a power rule?

The power of a power rule says that when you raise a power to another power, you multiply the two exponents together. For example, (2³)⁵ equals 2¹⁵, because 3 multiplied by 5 is 15. The base stays the same throughout.

Why do you multiply the powers in the power of a power rule?

Raising 2³ to the power of 5 means five sets of 2³ multiplied together. Each 2³ is three twos, so five sets give fifteen twos in total. That is why you multiply the exponents, giving 2¹⁵ rather than counting every two.

Why can you swap the exponents in the power of a power rule?

You can swap the two exponents because multiplication can be done in any order, known as the commutative property. So (aᵐ)ⁿ equals (aⁿ)ᵐ. Swapping can make a calculation easier, for example by turning a tricky negative exponent into a simpler positive one first.

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