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Powers and Roots (Economics)

Powers

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.

Zero Powers

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

Positive Integer Powers

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.

Negative Integer Powers

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, $a^b=\frac{1}{a^{(-b)}}.$

Fractional Powers
Positive Fractional Powers

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).

Negative Fractional Powers

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 $-b$.

Roots

Square Roots

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 \[\sqrt{(-25)}=\sqrt{(-1)}\times\sqrt{25}=5i.\]

Cube Roots

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.

Generalisation of Roots

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.

Exponent Laws

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.

Rule 1

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 \[a^{(-n)}\times a^{(-m)}=a^{(-n-m)}\]

Rule 2

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: \[\frac{a^n}{a^{(-m)}}=a^{n+m}\]

Rule 3

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.

Test Yourself

Powers and indices

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