Negative Indices
Every question with a fully worked solution.
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Theory
A negative index means a reciprocal: \(a^{-n} = \dfrac{1}{a^n}\), and any non-zero number to the power zero is \(1\). This Year 10 maths page shows where these rules come from, how to evaluate negative powers as fractions and decimals, and how to use them with the other index laws to simplify algebraic terms and write answers with positive indices.
The zero index: for any number \(a \ne 0\), \(a^0 = 1\). It follows from the division law: \(\dfrac{a^3}{a^3} = a^{3 - 3} = a^0\), and anything divided by itself is \(1\).
A negative index means the reciprocal of the positive power: \(a^{-n} = \dfrac{1}{a^n}\). For example, \(\dfrac{a^2}{a^5} = a^{-3}\), and cancelling gives \(\dfrac{1}{a^3}\).
A negative index does not make the number negative. \(7^{-1} = \dfrac{1}{7}\), which is positive and less than \(1\).
The index belongs only to what it is attached to. In \(4m^{-2}\) only \(m\) has the negative index, so \(4m^{-2} = \dfrac{4}{m^2}\).
All the index laws still work with zero and negative indices.
For \(a \ne 0\):
A fraction to a negative power:
How to evaluate or simplify
- Combine powers of the same base first with the index laws, adding or subtracting the indices (with their signs).
- Replace any zero power with \(1\).
- Rewrite each negative power as a reciprocal with a positive index. For a fraction, flip it.
- Evaluate as a fraction, or as a decimal if asked.
Finding a negative index
- Write the denominator as a power of the base: \(\dfrac{1}{64} = \dfrac{1}{4^3}\).
- Move it to the numerator with a negative index: \(4^{-3}\).
Other questions
- Decimals: change to a fraction first: \(0.25^{-1} = \left(\dfrac{1}{4}\right)^{-1} = 4\).
- Fractions of sums (complex fractions): multiply the top and bottom by the lowest common denominator; factorise if a difference of squares appears.
- Formulas: in \(P = P_0 \times 2^{-t}\), a negative \(t\) means a time in the past.
(i) \(4^{-2}\)
(ii) \(5^{-3}\), as a fraction and as a decimal
Write each with a positive index: \(a^{-n} = \dfrac{1}{a^n}\).
| \(4^{-2}\) | \(=\) | \(\dfrac{1}{4^2} = \boldsymbol{\dfrac{1}{16}}\) |
| \(5^{-3}\) | \(=\) | \(\dfrac{1}{5^3} = \boldsymbol{\dfrac{1}{125}}\) |
| \(\) | \(=\) | \(\boldsymbol{0.008}\) |
A negative index gives a small positive number, never a negative one.
(i) \(\left(\dfrac{2}{3}\right)^{-2}\)
(ii) \(\left(\dfrac{x^2}{5}\right)^{-2}\)
Take the reciprocal, then raise the top and bottom to the positive power.
| \(\left(\dfrac{2}{3}\right)^{-2}\) | \(=\) | \(\left(\dfrac{3}{2}\right)^2 = \boldsymbol{\dfrac{9}{4}}\) |
| \(\left(\dfrac{x^2}{5}\right)^{-2}\) | \(=\) | \(\left(\dfrac{5}{x^2}\right)^2\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{25}{x^4}}\) |
In (ii), square both parts: \(5^2 = 25\) and \((x^2)^2 = x^4\).
Raise every factor to the power \(-2\) (multiply each index by \(-2\)), then move negative powers below the line.
| \(\left(-3a^{2}b^{-3}\right)^{-2}\) | \(=\) | \((-3)^{-2}\,a^{-4}\,b^{6}\) |
| \(\) | \(=\) | \(\dfrac{1}{9} \times \dfrac{1}{a^4} \times b^6\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{b^6}{9a^4}}\) |
\((-3)^{-2} = \dfrac{1}{(-3)^2} = \dfrac{1}{9}\): the even power makes it positive.
(ii) Find \(n\) if \(3^n = \dfrac{1}{81}\).
(iii) Write \(6x^{-3}\) with a positive index.
The index laws work the same way with negative indices.
| \(10^4 \times 10^{-6}\) | \(=\) | \(10^{4 + (-6)} = 10^{-2}\) |
| \(\) | \(=\) | \(\dfrac{1}{100} = \boldsymbol{0.01}\) |
(ii) \(81 = 3^4\), so \(\dfrac{1}{81} = 3^{-4}\) and \(n = \boldsymbol{-4}\).
(iii) Only \(x\) has the negative index, so only \(x\) moves to the denominator: \(6x^{-3} = \boldsymbol{\dfrac{6}{x^3}}\).
Common pitfalls
Frequently asked questions
Why is any number to the power of 0 equal to 1?
Because \(\dfrac{a^n}{a^n} = a^0\) by the division law, and any non-zero number divided by itself is \(1\). The pattern \(8, 4, 2, 1\) for \(2^3, 2^2, 2^1, 2^0\) shows the same thing.
What does a negative index mean?
It means one over the positive power, the reciprocal: \(4^{-2} = \dfrac{1}{4^2} = \dfrac{1}{16}\).
Does a negative index make a number negative?
No. A negative index makes a reciprocal, which is still positive for a positive base: \(7^{-2} = \dfrac{1}{49}\).
How do you write a negative power as a decimal?
Write it as a fraction with a positive index, then divide: \(5^{-3} = \dfrac{1}{125} = 0.008\).
What is a fraction to a negative power?
Flip the fraction and make the power positive: \(\left(\dfrac{2}{3}\right)^{-2} = \left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\).