Power of a Power
The power of a power law says that raising a power to another power multiplies the indices, \((a^m)^n=a^{mn}\), and a power of a product or a fraction applies to every factor, as in \(\left(-\frac{2x^3}{5y}\right)^2=\frac{4x^6}{25y^2}\). In NSW Year 10 Advanced maths these laws are combined with multiplication and division (NESA outcome MA5‑IND‑C‑01).
You will simplify expressions with negative coefficients and brackets raised to powers, then multiply and divide the results. You will write numbers such as \(9^4\times27^2\div3^5\) with a common base, evaluate expressions without a calculator, find unknown indices, and work out the volume of a prism whose sides are powers of brackets. Harder questions prove an identity for every whole number \(n\) and count the small cubes inside a large cube. It is part of Year 10 Advanced Indices.
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: \((a^m)^n = a^{mn}\). Every factor inside a bracket is raised to the power, so \((ab)^n = a^nb^n\). This Year 10 Advanced maths page covers fractions and negative bases, writing numbers with a common base, and solving equations for an unknown index.
\((a^m)^n\) means \(n\) lots of \(a^m\) multiplied together. That is \(m \times n\) factors of \(a\), so \((a^m)^n = a^{mn}\).
A power applies to every factor inside the brackets: the coefficient, each pronumeral and the sign. So \((3x^2y)^2 = 3^2 \times (x^2)^2 \times y^2 = 9x^4y^2\). For a fraction, raise the numerator and the denominator.
An even power of a negative is positive; an odd power keeps the negative sign.
The law does not apply across addition: \((a + b)^2\) is not \(a^2 + b^2\).
| Law | In symbols | Example |
|---|---|---|
| Power of a power | \((a^m)^n = a^{mn}\) | \((p^4)^3 = p^{12}\) |
| Power of a product | \((ab)^n = a^nb^n\) | \((5k^2)^2 = 25k^4\) |
| Power of a quotient | \(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\) | \(\left(\dfrac{c}{4}\right)^3 = \dfrac{c^3}{64}\) |
| Negative base | \((-a)^n = a^n\) if \(n\) even, \(-a^n\) if \(n\) odd | \((-y^2)^5 = -y^{10}\) |
How to simplify
- Expand each bracket: raise the coefficient (with its sign) and multiply every index by the power outside.
- Multiply or divide the terms: coefficients as numbers, indices added or subtracted.
- Collect like terms only at the end, once they have the same pronumerals and indices (\(8x^6 + 3x^6 = 11x^6\)).
Other questions
- Evaluate without a calculator: write each number in prime-power form, such as \(12^4 = 2^8 \times 3^4\), then cancel.
- Unknown index: write both sides as powers of one base and equate the indices. Two unknowns need two equations, one for each base.
- Single power: \(4^x \times 8^{x}\) becomes \(2^{2x} \times 2^{3x} = 2^{5x}\).
- In terms of another pronumeral: if \(p = 3^a\), then \(9^{a} = (3^a)^2 = p^2\).
- Show that questions: simplify the left side step by step until it matches the right side.
- Index equations with \(n^2\): \((x^n)^n = x^{n^2}\); the equation may be a quadratic.
- Measurements: a volume or count such as \((2^4)^3\) small cubes uses the same law.
Raise every factor in each bracket to the power, including the number and its sign.
| \((-3x^3y)^2\) | \(=\) | \(9x^6y^2\) |
| \((2xy^4)^3\) | \(=\) | \(8x^3y^{12}\) |
| \(\text{numerator}\) | \(=\) | \(72x^9y^{14}\) |
| \(\dfrac{72x^9y^{14}}{12x^5y^9}\) | \(=\) | \(\boldsymbol{6x^4y^5}\) |
\((-3)^2 = 9\) is positive because the power is even.
Raise the numerator and the denominator to the power. An odd power of a negative is negative.
| \(\left(-\dfrac{2a^3}{3b^2}\right)^3\) | \(=\) | \(-\dfrac{2^3(a^3)^3}{3^3(b^2)^3}\) |
| \(\) | \(=\) | \(\boldsymbol{-\dfrac{8a^9}{27b^6}}\) |
With an even power, such as \(4\), the answer would be positive.
Write every number as a power of \(3\), using the power-of-a-power law.
| \(9^3\) | \(=\) | \((3^2)^3 = 3^6\) |
| \(27^n\) | \(=\) | \((3^3)^n = 3^{3n}\) |
| \(9^{13}\) | \(=\) | \((3^2)^{13} = 3^{26}\) |
| \(3^{6 + 3n - 1}\) | \(=\) | \(3^{26}\) |
| \(3n + 5\) | \(=\) | \(26\) |
| \(n\) | \(=\) | \(\boldsymbol{7}\) |
| \((x^n)^n\) | \(=\) | \(x^{n \times n} = x^{n^2}\) |
| \(x^{n^2 - 2n}\) | \(=\) | \(x^8\) |
| \(n^2 - 2n - 8\) | \(=\) | \(0\) |
| \((n - 4)(n + 2)\) | \(=\) | \(0\) |
| \(n\) | \(=\) | \(4 \text{ or } n = -2\) |
\(n\) must be positive, so \(n = \boldsymbol{4}\).
Check: \((x^4)^4 \div x^8 = x^{16} \div x^8 = x^8\) ✓
Common pitfalls
Frequently asked questions
What is the power of a power rule?
Multiply the indices: \((a^m)^n = a^{mn}\). For example, \((p^4)^3 = p^{12}\).
Do you raise the number in front to the power as well?
Yes. Every factor inside the brackets is raised to the power, so \((3x^2)^2 = 9x^4\).
What happens to a negative sign inside the brackets?
An even power makes the result positive and an odd power keeps it negative.
How do you solve an equation like 4 to the x equals 2 to the 10?
Write both sides with the same base: \(4^x = 2^{2x}\). Then \(2x = 10\), so \(x = 5\).
Is (a + b) squared equal to a squared plus b squared?
No. The index laws work for products, not sums: \((a + b)^2 = a^2 + 2ab + b^2\).