Exponents and Powers

1. EXPONENTS
In general if ‘x’ is any number and ‘n’ is any natural number, then we have ${\mathrm{x}}^{\mathrm{n}}=\mathrm{x}\times \mathrm{x}\times \mathrm{x}\dots \dots$. 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 $a^m\times a^n=a^{m+n}$

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

Quotient of powers of the same base $\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}=\left\{\begin{array}{l}{\mathrm{a}}^{\mathrm{m}-\mathrm{n}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }\mathrm{m}>\mathrm{n}\\ \frac{1}{{\mathrm{a}}^{\mathrm{m}-\mathrm{n}}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }\mathrm{n}>\mathrm{m}\end{array}\right.$

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) $\frac{a^m}{a^n}=\left\{\begin{array}{l}{a}^{m-n}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }m>n\\ \frac{1}{a^{m-n}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }n>m\end{array}\right.$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:

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

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.

$\mathrm{So},56750=5.6750\times {10}^4$