# Exponents and Powers

1. EXPONENTS
In general if ‘x’ is any number and ‘n’ is any natural number, then we have ${\mathrm{x}}^{\mathrm{n}}=\mathrm{x}×\mathrm{x}×\mathrm{x}\dots \dots$. n times. The number ‘x’ is called the base and ‘n’ is called the exponent (or) the index of the exponential expression, ${x}^{n}$.
Eg: In exponential form, we write 2 × 2 × 2 as ${2}^{3}$ read as 2 raised to the power 3.
We have, base = 2 and exponent = 3
Rule
If $\frac{p}{q}$ is any fractional number, then for any positive integer ‘m’, we have ${\left(\frac{\mathrm{p}}{\mathrm{q}}\right)}^{\mathrm{m}}=\frac{{\mathrm{p}}^{\mathrm{m}}}{{\mathrm{q}}^{\mathrm{m}}}$.

Reciprocal
The reciprocal of a non zero integer ‘x’ is denoted by ${x}^{–1}$ and defined as ${\mathrm{x}}^{-1}=\frac{1}{\mathrm{x}}$.

For a fractional number $\frac{p}{q}$ (where $p\ne 0;q\ne 0$) we have ${\left(\frac{\mathrm{p}}{\mathrm{q}}\right)}^{-1}=\frac{\mathrm{q}}{\mathrm{p}}$.

The reciprocal of ${\left(\frac{p}{q}\right)}^{m}$is given by ${\left(\frac{\mathrm{q}}{\mathrm{p}}\right)}^{\mathrm{m}}$.

2. LAWS OF EXPONENTS
Law 1

The product of the two powers of the same base is a power of the same base with the
index equal to the sum of the indices.
i.e., if $a\ne 0$ be any rational number and m, n be positive integers, then ${a}^{m}×{a}^{n}={a}^{m+n}$

Law 2

Power of a power. i.e.,${\left({a}^{m}\right)}^{n}={a}^{mn}$ for all positive integers
Law 3

Power of a product $\left(ab{\right)}^{m}={a}^{m}×{b}^{m}$, where $a\ne 0,b\ne 0$ and ‘m’ is a positive integer.
Repeated application of this gives a more general result namely $\left(abc..z{\right)}^{m}={a}^{m}{b}^{m}{c}^{m}\dots \dots ..{Z}^{m}$
Law 4

Quotient of powers of the same base

Law 5

power of a Quotient i.e.,  ${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}$ where $a\ne 0,b\ne 0$ and ‘m’ is a positive integer.

POWERS WITH ZERO AND NEGATIVE EXPONENTS
i) ${a}^{0}=1$ for every non zero real number ‘a’
ii) where m, n are positive integers and $a\ne 0$ . If we denote the multiplicative inverse of an by ${a}^{-n}\cdot {a}^{-n}=\frac{1}{{a}^{n}}$

EXPRESSING LARGE NUMBERS IN THE STANDARD FORM
In earlier classes, we have learnt to write a number in the expanded form, as shown below:
473 = 4 ×100 + 7 × 10 + 3
3758 = 3 × 1000 + 7 × 100 + 5 × 10 + 8
30739 = 3 × 10000 + 0 × 1000 + 7 × 100 + 3 × 10 + 9
We can express these using powers of 10 in the exponential form:

$\begin{array}{l}473=4×{10}^{2}+7×{10}^{1}+3×{10}^{0}\\ 3758=3×{10}^{3}+7×{10}^{2}+5×{10}^{1}+8×{10}^{0}\\ 30739=3×{10}^{4}+0×{10}^{3}+7×{10}^{2}+3×{10}^{1}+9×{10}^{0}\end{array}$

In this section, we shall learn to write large numbers, using powers of 10 as shown above.
Standard Form Any number can b written as a number between 1 and 10 multiplied by a power of 10. This is called standard form of the number.
To write a number in standard form we split it into two parts multiplied together. The first part must be a number between 1 and 10 and the second, a power of 10.
For the number 56750, we start with 5.6750 in order to have a number between 1 and 10; and then we move the decimal point to the right until it is in its correct place (i.e., 56750.0).
Here , it needs to move 4 places.
The number 4 is the power of 10.

$\mathrm{So},56750=5.6750×{10}^{4}$