Vachmi
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
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
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.
These are called negative exponents.
($x^{-m}$) = ($1/x^{m}$)
e.g. ($2^{-5}$) = ($1/2^{5}$) = ($1/32$) = 0.03125
$(x/y)^{-m}$ = $(y/x)^{m}$
e.g. ($2^{-5}$) = ($1/2^{5}$) = ($1/32$) = 0.03125
$(x/y)^{-m}$ = $(y/x)^{m}$
Please refer Solved Examples here.
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 |
