Power of a Power
Every question with a fully worked solution.
- Power of a Power - Video - Indices - Bracket to a Power Watch
Theory
To raise a power to a power, multiply the indices: \((a^m)^n = a^{mn}\). This Year 10 maths page also shows how to raise a product to a power, handle negative coefficients, combine all the index laws, and write numbers such as \(8\) and \(16\) with a common base.
A power of a power such as \((a^4)^3\) means \(a^4 \times a^4 \times a^4\). That is \(3\) groups of \(4\) factors, so multiply the indices: \(a^{12}\).
A power of a product such as \((5xy^2)^2\) raises every factor inside the brackets to the power: \(5^2x^2y^4 = 25x^2y^4\). The number in front is raised to the power too.
With a negative number inside, the sign depends on the power. An even power gives a positive result; an odd power gives a negative result.
To combine the laws, remove brackets first, then multiply (add indices) and divide (subtract indices).
Numbers can be rewritten with a common base: \(4 = 2^2\), \(8 = 2^3\), \(16 = 2^4\), \(9 = 3^2\), \(27 = 3^3\), \(25 = 5^2\).
For positive integers \(m\) and \(n\):
With a coefficient \(p\):
How to simplify a power of a power
- Raise every factor inside the brackets to the power: the number, and each pronumeral.
- Work out the number: \(2^4 = 16\), \((-3)^2 = 9\). Check the sign for a negative number.
- Multiply the indices of each pronumeral.
- Finish any multiplying or dividing that is left, with the other index laws.
Different bases
- Rewrite each number as a power of the same base, such as \(2\) or \(3\).
- Apply the power-of-a-power law, then add or subtract indices.
(i) \((7^3)^4\), as a single power of \(7\)
(ii) \((2y^5)^3\)
Multiply the indices. In (ii), every factor inside the brackets is raised to the power, including the \(2\).
| \((7^3)^4\) | \(=\) | \(7^{3 \times 4} = \boldsymbol{7^{12}}\) |
| \((2y^5)^3\) | \(=\) | \(2^3 \times y^{5 \times 3}\) |
| \(\) | \(=\) | \(\boldsymbol{8y^{15}}\) |
\(2^3 = 8\), not \(6\).
(i) \((-2a^4)^3\)
(ii) \((-5m^2)^2\)
An odd power of a negative number is negative. An even power is positive.
| \((-2a^4)^3\) | \(=\) | \((-2)^3 \times a^{12}\) |
| \(\) | \(=\) | \(\boldsymbol{-8a^{12}}\) |
| \((-5m^2)^2\) | \(=\) | \((-5)^2 \times m^{4}\) |
| \(\) | \(=\) | \(\boldsymbol{25m^4}\) |
Remove the brackets first with the power-of-a-power law. Then divide: divide the numbers and subtract the indices.
| \((2a^3b^2)^4\) | \(=\) | \(2^4a^{12}b^8 = 16a^{12}b^8\) |
| \(\dfrac{16a^{12}b^8}{8a^4b^3}\) | \(=\) | \(2a^{12 - 4}b^{8 - 3}\) |
| \(\) | \(=\) | \(\boldsymbol{2a^8b^5}\) |
Write each number as a power of \(3\): \(9 = 3^2\) and \(27 = 3^3\). Then use both index laws.
| \(9^2 \times 27\) | \(=\) | \((3^2)^2 \times 3^3\) |
| \(\) | \(=\) | \(3^4 \times 3^3\) |
| \(\) | \(=\) | \(\boldsymbol{3^7}\) |
Check: \(81 \times 27 = 2187 = 3^7\).
Common pitfalls
Frequently asked questions
What is the power of a power rule?
When a power is raised to another power, multiply the indices: \((a^m)^n = a^{mn}\). For example, \((x^4)^3 = x^{12}\).
Do you add or multiply the indices?
Add them when multiplying two powers of the same base, such as \(x^2 \times x^3 = x^5\). Multiply them when a power is raised to another power, such as \((x^2)^3 = x^6\).
What happens to the number in front when you raise brackets to a power?
The number is raised to the power as well: \((2y^5)^3 = 2^3y^{15} = 8y^{15}\).
Is a negative number to an even power positive?
Yes. Pairs of negatives multiply to give positives, so an even power is positive and an odd power is negative: \((-2)^4 = 16\) but \((-2)^3 = -8\).
How do you write 49 cubed as a power of 7?
Write \(49\) as \(7^2\), then multiply the indices: \((7^2)^3 = 7^6\).