Powers and roots

Put simply, raising to a power is a mathematical operation used as a shorthand for repeated multiplication.

Positive integer indices

A power has the form aⁿ, where n is called the index (or exponent) and a is called the base. What interests us most from here on is the kind of index. In this section we look at powers with a positive integer index, that is, when the index is one of the numbers 1, 2, 3, 4, …

An example of such a power is 6². The 6 is the base and the 2 is the index. The index tells us how many sixes we multiply together to get the result. So 6² = 6 · 6 = 36. Another example is 4³ = 4 · 4 · 4 = 64. In general we can write it like this:

$a^{n} = \underset{n times}{\underset{⏟}{a \cdot a \cdot \dots \cdot a}}$

The base can be anything – a number, or even a more complicated expression. In the calculation you simply multiply together as many copies of that expression as the index says. Examples:

$\begin{array}{rcl} (- 5)^{3} & = & (- 5) \cdot (- 5) \cdot (- 5) \\ x^{4} & = & x \cdot x \cdot x \cdot x \\ (x + 3)^{2} & = & (x + 3) \cdot (x + 3) \end{array}$

For an index equal to zero we define a⁰ = 1.

Negative indices

In this section we assume that the index is negative. What could that mean? We start from a⁰ = 1. To get a¹, we can say that we multiply a⁰ by a. Since a⁰ = 1, the product is a⁰ · a = a. We get a. To get a², we can write a² = a⁰ · a · a.

We can think of a⁰ = 1 as our starting value: to work out aⁿ, we just multiply a⁰ by a a total of n times. How would we proceed if n were negative? If n is positive, we multiply. If n is negative, we divide. So to get a⁻¹ we take the starting value a⁰ and divide it by a. This gives the equation:

$a^{- 1} = \frac{a^{0}}{a} = \frac{1}{a}$

If we wanted a⁻², we would divide a⁰ by a twice. What would we get? First let’s look at another way of writing division. If we have the quotient x/y, we can just as well write it as

$x \cdot \frac{1}{y}$

This is one of the basic properties of fractions. So if we want to divide an expression x by an expression y twice, we can write it as

$x \cdot \frac{1}{y} \cdot \frac{1}{y}$

Let’s go back to the example a⁻². We said that we get it by dividing a⁰ by a twice. We can rewrite that like this:

$a^{- 2} = a^{0} \cdot \frac{1}{a} \cdot \frac{1}{a} = a^{0} \frac{1}{a \cdot a} = 1 \cdot \frac{1}{a^{2}} = \frac{1}{a^{2}}$

Now we have a method for working out a power with a negative index. We work out the power as if the index were positive and then take the reciprocal, that is, divide 1 by the result. Written more precisely:

$a^{- n} = \frac{1}{a^{n}}$

(Here we assume that −n is negative, so n is positive.)

Some examples:

$\begin{array}{rcl} 2^{- 1} & = & \frac{1}{2^{1}} = \frac{1}{2} \\ 5^{- 3} & = & \frac{1}{5^{3}} = \frac{1}{125} \\ (2 x + 3)^{- 8} & = & \frac{1}{(2 x + 3)^{8}} \end{array}$

Fractional indices

We can extend the index further to all rational numbers. A rational number is a number that can be written as a fraction, the quotient of two integers. Take a fraction m/n, where n is positive. Then we can write the formula

$a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$

The funny-looking sign over a^m is the symbol for a root.

Calculator

If you need to work out a power, you can use our powers calculator 🧮.

Laws of indices

a⁰ = 1, if a ≠ 0.
a¹ = a.
0ⁿ = 0, if n > 0.
0⁰ is undefined.
(a · b)ⁿ = aⁿ · bⁿ.
a^m · aⁿ = a^{m + n}.
$\frac{a^{m}}{a^{n}} = a^{m - n}$ , if a ≠ 0.
(a^m)ⁿ = a^{m · n}.

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