Powers (also called **exponents** or **indices**) are a fast and tidy way to express multiplications that have many repeated numbers.

For instance, if $a$ is a number (or variable) and we have $a\times a\times a \times$ then we write this as: \[a^4\] which is pronounced as *“$a$ to the power of four”* as there are *four* lots of $a$ multiplied together. In general, if we have $b$ lots of $a$ multiplied together, then we write this as: \[a^b\] This is pronounced as “$a$ raised to the power of $b$”. The number (or variable) $b$ is referred to as the **power, index** or **exponent** and $a$ is referred to as the **base**.

See Powers for more information about powers.

Any number raised to the power of zero is equal to $1$.

When $b$ is a positive integer, $a^b$ means that we have $b$ lots of $a$ multiplied together.

To **“square”** a number, we multiply the same number by itself *once*. The result is a **square number**. Thus “five squared” means $5\times5$ and is written as $5^2$. We have $5^2=5\times5=25$ so $25$ is a square number.

$7\times7\times7$ means “seven **cubed**” and is written as $7^3$. We have $7^3=7\times7\times7=343$.

**Note**: A negative number raised to the power of an even number will always be *even*. This is because when two negative numbers are multiplied by each other the two minus signs ‘cancel out’ and the result is a positive number. For example, $(-7)^2=49$ and $7^2=49$. If a negative number is raised to the power of an odd number, the result will always be *negative*.

We can also raise a number to a *negative power*. Raising the numner $a$ to the negative power $-b$ is equivalent to dividing $1$ by $a$ raised to the equivalent positive power, $b$. That is, \[a^{(-b)}=\frac{1}{a^b}\] For example, \[5^{(-2)}=\frac{1}{5^2}=\frac{1}{25}=0.04.\]

**Note**: we can write positive integer powers as $1$ divided by the number raised to the negative power. That is, $a^b=\frac{1}{a^{(-b)}}.$

Raising the number $a$ to the power of $\frac{1}{b}$ is equivalent to taking the $b$th root of $a$. That is, \[a^\left(\frac{1}{b}\right)=\sqrt[b]{a}\]

Raising the number a to the power of $\frac{b}{c}$ is equivalent to first raising $a$ to the power of $b$ and then taking the $c$th root. That is, \[a^{\left(\frac{b}{c}\right)}=\sqrt[c]{a^b}.\]

**Note:** We would obtain the same result by first taking the $c$th root of $a$ and then raising the result to the power of $b$ (try it yourself).

Raising the number $a$ to the power of $\frac{–b}{c}$ is equal to raising $a$ to the power of $–b$ and then taking the $c$th root of the result: \[a^{\left(\frac{-b}{c}\right)}=\frac{1}{\sqrt[c]{a^b}}.\]

**Note**: As for positive fractional powers, we would obtain the same result by first taking the $c$th root of $a$ and then raising the result to the power of $-b$.

The opposite of a square number is a **square root**. We denote the square root of the number $a$ by $\sqrt{a}$.Taking the square root of a square number $b^2$ is thus equal to the number itself, $b$.

For example, $7^2=49$ so $\sqrt{49}=\sqrt{7^2}=7$.

**Note**: Recall from above that the square of a negative number is the same as the square of the same positive number. Using the example given above, both the square of $7$ and the square of $(-7)$ are equal to $49$. However, only $7$ is the square root of $49$; $(-7)$ is equal to the *negative* of the square root, $-\sqrt{49}$. We write this as \[\pm\sqrt{49}=\pm7.\]

**Note**: The square root of a negative number is an imaginary number. That is, we have \[\sqrt{(-25)}=\sqrt{(-1)}\times\sqrt{25}=5i.\]

The opposite of a cube number is a **cube root**. We denote the cube root of the number $8$ by $\sqrt[3]{8}$ so we have $\sqrt[3]{8}=2$. The cube root of any positive number is also positive.

We denote the $nth$ root of the number $a$ by $\sqrt[n]{a}$ or (less commonly) by $a^\left(\frac{1}{n}\right)$. This is the number which, when multiplied by itself $n$ times, is equal to the number $a$.

**Note**: As $\sqrt[n]{a}$ is the same thing as $a^\left(\frac{1}{n}\right)$ for any numbers $n$ and $a$ we can see that **all roots can be written as fractional powers**.

**Note**: When $n$ is odd (see cubed roots above) the root can only be *positive*, while when $n$ is even (see square roots above) the root can be either *positive or negative*.

There are four rules which we can use to simplify expressions involving arithmetic with exponents.

**Note**: Since we've seen that roots are equivalent to fractional powers, these laws can also be applied to arithmetic with roots.

When we multiply together two numbers with the *same base* and *positive* exponents, we *add* their exponents: \[a^n\times a^m=a^{n+m}\] When we multiply together two numbers with the same base and one or both has a negative exponent, we *subtract* all negative exponent(s): \[a^n\times a^{(-m)}=a^{n-m}\] or \[a^{(-n)}\times a^{(-m)}=a^{(-n-m)}\]

When we divide one number by another with the *same base* and both numbers have *positive* exponents, we *subtract* the exponent of the denominator from the exponent of the numerator: \[\frac{a^n}{a^m}=a^{n-m}\] When we divide one number by another with the *same base* and the denominator has a *negative* exponent, we *add* the exponents: \[\frac{a^n}{a^{(-m)}}=a^{n+m}\]

When we raise the number $a^b$ to the power $c$, we multiply the number $b$ by $c$: \[(a^b)^c\]

**Note**: $a$, $b$ and $c$ can each be either positive or negative.