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: \((a^m)^n = a^{mn}\). Every factor inside a bracket is raised to the power, so \((ab)^n = a^nb^n\). This Year 10A maths page covers powers of products and fractions, negative bases, combining all the index laws, writing numbers with a common base and solving equations for an unknown index.
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.
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.
- Measurements: a volume or count such as \((2^4)^3\) small cubes uses the same law.
(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 \(6\).
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.
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.
Write every number as a power of \(3\), using the power-of-a-power law.
| \(9^3\) | \(=\) | \((3^2)^3 = 3^6\) |
| \(27^n\) | \(=\) | \((3^3)^n = 3^{3n}\) |
| \(9^{13}\) | \(=\) | \((3^2)^{13} = 3^{26}\) |
| \(3^{6 + 3n - 1}\) | \(=\) | \(3^{26}\) |
| \(3n + 5\) | \(=\) | \(26\) |
| \(n\) | \(=\) | \(\boldsymbol{7}\) |
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: \((7^2)^3 = 7^6\).
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 \(x = 5\).