Power of a Power
To raise a power to another power, multiply the indices: \((a^m)^n = a^{mn}\). Every factor inside the bracket is raised to the power, so \((3x)^2 = 9x^2\). This NSW Year 9 Advanced maths subtopic practises the rule with algebraic terms (NESA outcome MA5‑IND‑C‑01).
You will simplify expressions such as \((-3m^2)^4\) and \(\left(\frac{2t}{r^2}\right)^3\), watch the sign when a negative base has an odd or even power, and find \(a\) and \(b\) in \((-3x^5)^4 = ax^b\). You will write \(9^5 \times 27^3\) as a power of \(3\), find a cube's side and surface area from its volume, and decide whether \(2^{30}\) or \(3^{20}\) is larger. It is part of Year 9 Advanced Indices and Scientific Notation.
Every question with a fully worked solution.
- Power of a Power - Video - Index Law Watch
Theory
To raise a power to a power, multiply the indices: \((a^m)^n = a^{mn}\). This Year 9 Advanced maths page also covers raising every factor in a bracket, fractions in brackets, negative signs with odd and even powers, rewriting bases and finding an unknown index.
A power of a power such as \((a^2)^3\) means three copies of \(a^2\) multiplied together. That is \(2 \times 3 = 6\) factors of \(a\), so the indices are multiplied.
When a bracket holds a product, the index outside applies to every factor inside: \((2b)^3 = 2^3b^3 = 8b^3\). The coefficient is raised too, and it is raised to the power, not multiplied by it.
When a bracket holds a fraction, the top and the bottom are both raised: \(\left(\dfrac{a}{b}\right)^n\) \(= \dfrac{a^n}{b^n}\).
A negative number to an even power is positive; to an odd power it is negative. So \((-1)^6 = 1\) and \((-1)^7 = -1\).
The law also runs backwards. \(3^{12}\) can be written \((3^2)^6 = 9^6\), \((3^3)^4 = 27^4\) or \((3^4)^3 = 81^3\). This is how you match bases and compare large powers.
| Expression | Working | Result |
|---|---|---|
| \((p^5)^3\) | \(5 \times 3\) | \(p^{15}\) |
| \((5c^2d)^3\) | \(5^3c^{2 \times 3}d^{3}\) | \(125c^6d^3\) |
| \((-k^3)^4\) | even power: sign lost | \(k^{12}\) |
| \((-k^3)^5\) | odd power: sign kept | \(-k^{15}\) |
| \(\left(\dfrac{w^3}{2}\right)^5\) | \(\dfrac{w^{15}}{2^5}\) | \(\dfrac{w^{15}}{32}\) |
| \(\big((m^2)^3\big)^2\) | \(2 \times 3 \times 2\) | \(m^{12}\) |
How to raise a bracket to a power
- List the factors inside the bracket: the number (with its sign) and each pronumeral.
- Number: raise it to the power. Decide the sign first: odd power keeps a minus, even power removes it.
- Pronumerals: multiply each index by the outside index. A pronumeral with no index shown has index \(1\).
- Fractions: raise the top and the bottom separately.
- Nested brackets: multiply all the outer indices together, then raise each factor.
Problems built on the law
- Mixed with \(\times\) and \(\div\): expand every bracket first, then add or subtract indices.
- Find a coefficient and index: if \((-4x^3)^2 = ax^b\), expand the left side and match: the number part gives \(a\), the index gives \(b\).
- Different bases: rewrite each base as a power of the same prime, then equate the indices (Example 2).
- Comparing large powers: rewrite both with the same index, then compare the bases (Example 3).
- Length from area or volume: a square of area \(A\) has side \(\sqrt{A}\); a cube of volume \(V\) has side \(\sqrt[3]{V}\). Halve or third each index.
(i) \((-2a^4b^3)^5\)
(ii) \(\left(-\dfrac{3x}{y^2}\right)^4\).
Raise every factor inside the bracket. Count the minus signs: an odd power keeps the sign, an even power loses it.
(i)
| \((-2a^4b^3)^5\) | \(=\) | \((-2)^5 \times a^{4 \times 5} \times b^{3 \times 5}\) |
| \(\) | \(=\) | \(\boldsymbol{-32a^{20}b^{15}}\) |
(ii)
| \(\left(-\dfrac{3x}{y^2}\right)^4\) | \(=\) | \(\dfrac{(-3)^4x^4}{y^{2 \times 4}}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{81x^4}{y^8}}\) |
\((-2)^5 = -32\) because five negatives multiply to a negative. \((-3)^4 = 81\) because four negatives pair up.
Write every base as a power of \(5\), then use the laws.
| \(25^4\) | \(=\) | \((5^2)^4 = 5^8\) |
| \(125\) | \(=\) | \(5^3\) |
| \(\dfrac{5^8 \times 5^3}{5^3}\) | \(=\) | \(5^{8+3-3} = 5^8\) |
| \(\therefore\ n\) | \(=\) | \(\mathbf{8}\) |
Learn the small powers that come up most: \(4 = 2^2\), \(8 = 2^3\), \(9 = 3^2\), \(27 = 3^3\), \(25 = 5^2\), \(125 = 5^3\).
The indices \(30\) and \(70\) share the factor \(10\). Rewrite both as tenth powers.
| \(5^{30}\) | \(=\) | \((5^3)^{10} = 125^{10}\) |
| \(2^{70}\) | \(=\) | \((2^7)^{10} = 128^{10}\) |
Both are now tenth powers of a positive number, and \(128 > 125\), so \(128^{10} > 125^{10}\).
| \(\therefore\ 2^{70}\) | \(>\) | \(5^{30}\) |
\(2^{70}\) is larger, even though its base is smaller. Comparing bases alone, or indices alone, does not decide it.
Multiply out each power of a power, then combine with the multiplication and division laws.
| \((x^{k})^4\) | \(=\) | \(x^{4k}\) |
| \((x^3)^{k}\) | \(=\) | \(x^{3k}\) |
| \(\dfrac{x^{4k} \times x^3}{x^{3k}}\) | \(=\) | \(x^{4k+3-3k} = x^{k+3}\) |
The bases match, so the indices are equal:
| \(k + 3\) | \(=\) | \(10\) |
| \(k\) | \(=\) | \(\mathbf{7}\) |
Check: \(\dfrac{x^{28} \times x^3}{x^{21}}\) \(= x^{31-21}\) \(= x^{10}\).
Common pitfalls
Frequently asked questions
What is the power of a power rule?
When a power is raised to another power, multiply the indices: \((a^m)^n = a^{mn}\).
Do you raise the number in front to the power as well?
Yes, if it is inside the bracket. Every factor inside is raised, so \((3y)^2 = 9y^2\). A number outside the bracket is not raised.
What happens to a negative sign inside a bracket raised to a power?
An even power makes the result positive and an odd power keeps it negative: \((-2)^4 = 16\) and \((-2)^3 = -8\).
How do you compare two large powers without a calculator?
Rewrite them with the same index using the power of a power law, for example as tenth powers. Then the one with the larger base is larger.
How do you raise a fraction to a power?
Raise the numerator and the denominator separately: \(\left(\dfrac{a}{b}\right)^n\) \(= \dfrac{a^n}{b^n}\).