### What are Exponents?

Exponents are repeated multiplication of the same number by itself.

For example, to multiply 4 three times by itself, we write it as 4 * 4 * 4.

The above multiplication can also be written as $4^3$ and is same as = 4 * 4 * 4 = 64

In this example, 4 is called the "base" and 3 is called the "exponent".

This process is also called as "raising the base to a power of". In above example, 4 is raised to the power of 3.

If a number is raised to the power of 2, it is called the square of a number.

If a number is raised to the power of 3, it is called as cube of a number.

For higher powers, there are no specific names.

Let us try with some examples.

When we say $7^5$, it means 7 * 7 * 7 * 7 * 7 = 16807

$3 * 3 * 3 * 3 = 81$ is same as $3^4$ = 81

$(-7)^3$ means (-7) * (-7) * (-7) = -343

Now try to solve this yourself. $6^3$ = ?

Interestingly, any number raised to power 0 equals 1 e.g. $5^0 = 1$

Have you ever wondered why?

### Laws of Exponents

(i) When multiplying two terms with the same bases, we can add the exponents.

($x^m$) ($x^n$) = ($x^{m + n}$)

e.g.
($2^4$) ($2^3$) = ($2^{4 + 3}$) = ($2^{7}$) = 128

(ii) When raising an exponent term to a power, we can multiply the outer power by inner power.

$(x^m)^n$ = ($x^{m * n}$)

e.g.
$(3^2)^5$ = ($3^{2 * 5}$) = ($3^{10}$) = 59049

(iii) When raising a term to negative power, we can take reciprocal of the term.

($x^{-m}$) = ($1/x^{m}$)

e.g.
($2^{-5}$) = ($1/2^{5}$) = ($1/32$) = 0.03125

$(x/y)^{-m}$ = $(y/x)^{m}$

Solved Examples
### Exponential functions

Consider a function of the form f(x) = $a^x$, where a > 0.

The following example represents 4 graphs for various values of a.

$x$ |
$1/2^x$ |
$2^x$ |
$1/e^x$ |
$e^x$ |

-3 |
8 |
0.125 |
20.08553692 |
.0497870680 |

-2 |
4 |
0.25 |
7.389056099 |
0.135335283 |

-1 |
2 |
0.5 |
2.718281828 |
0.367879441 |

0 |
1 |
1 |
1 |
1 |

1 |
0.5 |
2 |
0.367879441 |
2.718281828 |

2 |
0.25 |
4 |
0.135335283 |
7.389056099 |

3 |
0.125 |
8 |
.0497870680 |
20.08553692 |