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# Exponents

### 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