Power of a Power
To raise a power to another power, keep the base and multiply the indices: \((a^m)^n = a^{mn}\). In NSW Year 8 maths this sits alongside the power of a product, where every factor inside the bracket is raised to the outside index (NESA outcome MA4‑IND‑C‑01).
You will simplify expressions such as \((3^2)^4\) and \((3 \times 2^4)^2\), deal with negative bases like \(((-2)^3)^3\), write \(25^4\) as a power of 5 and work backwards to a missing index. You will also extend the law to pronumerals, a first look at Stage 5 work, in \((3x^2y)^4\). The number of square metres in a square kilometre and the small cubes in a large cube are found with this law. It is part of Year 8 Algebra.
Every question with a fully worked solution.
- Power of a Power - Video - Power of a Power - Example 1 Watch
Theory
When raising a term with a power to another power, the indices are multiplied: \((a^m)^n = a^{m \times n} = a^{mn}\). Every factor inside the bracket is raised to the outside index: \((3a^2)^3 = 3^3 \times a^{2 \times 3} = 27a^6\). This Year 8 maths page shows how.
Power of a power. \((a^m)^n = a^{m \times n}\). The bracket means \(n\) lots of \(a^m\) multiplied, so the index \(m\) is added \(n\) times, which is \(m \times n\). The order does not matter: \((5^3)^2 = (5^2)^3 = 5^6\).
Power of a product. Every factor inside the bracket is raised: \((a \times b)^n = a^n \times b^n\). Check: \((3 \times 4)^2 = 144\) and \(3^2 \times 4^2 = 9 \times 16 = 144\). A number in front is raised too: \((3a^2)^3 = 27a^6\), not \(3a^6\).
Negative signs. A minus sign inside the bracket is a factor of \(-1\): an odd outside index gives a negative, an even one a positive. \((-2a^4)^3 = (-2)^3 \times a^{12} = -8a^{12}\).
Rewriting a base. \(27 = 3^3\), so \(27^5 = (3^3)^5 = 3^{15}\). To compare \(4^7\) and \(16^3\), write both as powers of \(2\): \(2^{14}\) and \(2^{12}\).
Working backwards. In \((2^{?})^4 = 2^{20}\) the missing index is \(20 \div 4 = 5\). Do the bracket first, then use the multiplication or division law.
Raising to a power:
How to simplify a power of a power
- Raise the number in front to the outside index (watch its sign).
- Multiply each pronumeral’s index by the outside index.
- Then use the multiplication or division law if needed.
| \((3a^2)^3\) | \(=\) | \(3^3 \times a^{2 \times 3}\) |
| \(\) | \(=\) | \(27a^6\) |
\(27a^6\)
| \((-2a^4)^3\) | \(=\) | \((-2)^3 \times a^{4 \times 3}\) |
| \(\) | \(=\) | \(-8a^{12}\) |
\(-8a^{12}\) (odd index, negative)
(i) \((6^3)^4\)
(ii) \(\left((-4)^2\right)^5\)
| \((6^3)^4\) | \(=\) | \(6^{12}\) |
| \(\left((-4)^2\right)^5\) | \(=\) | \((-4)^{10}\) |
(i) \(6^{12}\)
(ii) \((-4)^{10}\), which is positive (even index)
| \((5 \times 2^3)^4\) | \(=\) | \(5^4 \times 2^{12}\) |
\(5^4 \times 2^{12}\). Check: \(40^4 = 2\,560\,000 = 625 \times 4096\).
(i) Write its area as a power of \(3\).
(ii) It is split into square plots of side \(3^2\) m. How many plots are there?
| \((3^4)^2\) | \(=\) | \(3^8\) |
| \(3^8 \div 3^4\) | \(=\) | \(3^4 = 81\) |
(i) \(3^8 = 6561\) m\(^2\)
(ii) Each plot is \((3^2)^2 = 3^4\) m\(^2\), so \(81\) plots
| \(4^7\) | \(=\) | \((2^2)^7 = 2^{14}\) |
| \(16^3\) | \(=\) | \((2^4)^3 = 2^{12}\) |
\(4^7\), because \(2^{14} > 2^{12}\) (\(16\,384 > 4096\))
Common pitfalls
Frequently asked questions
What is the power of a power law?
Multiply the indices: \((a^m)^n = a^{mn}\), so \((x^4)^3 = x^{12}\).
How do you raise a bracket with a number in it?
Raise every factor: \((2x^3)^4 = 2^4 x^{12} = 16x^{12}\).
Does the order of the indices matter?
No: \((5^3)^2\) and \((5^2)^3\) both equal \(5^6\).
What happens to a negative sign inside the bracket?
It is raised too: an odd outside index keeps it negative, an even one makes it positive.
How do you compare powers with different bases?
Rewrite them with the same base: \(9^3 = (3^2)^3 = 3^6\).