Indices

Indices are used to write products of repeated factors. For example, the product $3\times3\times3\times3\times3$ can be written as $3^5$ , where the number $5$ shows that $5$ factors of $3$ appear in the product.
Consider the following products:
$3\times3=3^2$
$3\times3\times3=3^3$
$3\times3\times3\times3=3^4$
. . .
. . .
. . .
$3\times3\times3\times...\times3(n-times)=3^n$
To generalise, the symbol $a^n$ is defined as follows:
If $n$ is a positive integer and $a$ real,
$a^n=a\times a\times a\times...\times a(n-times)=a^n$ . $a^0=1$ , $a neq 0$ . The integer $n$ is called the power or index or exponent and $a$ is called the base.

Laws of indices

Working with indices can greatly be simplified by using the various properties of indices. These properties include:

Multiplication

To multiply two numbers (say $a^m$ and $a^n$ where $m$ and $n$ are positive integers and $a$ is real) written in index notation with the same bases, we simply add the powers of the numbers.
ie $a^m\times a^n=a^{m+n}$

Examples

Simplify the following leaving your answer in index form.
1) $3^2\times3^4\times3^5=3^{2+4+5}$
$=3^{11}$
2) $16\times2^3\times2^5=2^4\times2^3\times2^5$
$=2^{4+3+5}$
$=2^{12}$

Division

To divide two numbers written in index form with the same bases, we subtract the exponent of the number in the denominator from the power of the number in the numerator.
ie $\frac{a^m}{a^n}=a^{m-n}$

Examples

Simplify the following:
1) $\frac{2^8}{2^5}=2^{8-5}$
$=2^3$
2) $\frac{m^4p^2}{m^3p^5}=m^{4-3}p^{2-5}$
$=mp^{-3}$

Powers of indices

Given any number in index form raised to any power, we simply multiply the exponent by the power.
ie $(a^m)^n=a^{mn}$
Example is $(3^5)^2=3^{5\times2}=3^{10}$

Negative exponent

Consider the following:
$\frac{5^2}{5^4}=5^{2-4}=5^{-2}$
Also,
$\frac{5^2}{5^4}=\frac{5\times5}{5\times5\times5\times5}=\frac{1}{5^2}$
we notice that $5^{-2}=\frac{1}{5^2}$
Generally, $a^{-n}=\frac{1}{a^n}$ .

Fractional indices

For all real numbers $a$ , a meaning has been given to $a^n$ for all real values of $n$ . Meaning can also be given to $a^n$ for rational $n$ .
For all integers $m$ and $n$ , and all positive integers $n$ such that $\frac{m}{n}$ , $n neq0$ is in its lowest terms, and all non-zero real $a$ ,
$a^{\frac{m}{n}}=\sqrt[n]{a^m}$ .

Summary

1) $a^m\times a^n= a^{m+n}$
2) $\frac{a^m}{a^n}=a^{m-n}$
3) $(a^m)^n=a^{mn}$
4) $a^{-n}=\frac{1}{a^n}$
5) $a^{\frac{m}{n}}=\sqrt[n]{a^m}$

Try these!

1) $2^3\times2^8\times2^{1-p}$
2) $(p^{5m})^3(p^3)^m$
3) $(2p^{2t})(p^{t+1})$
4) $a^{2t}(3+a^t)$
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Indices

Laws of indices

Multiplication

Examples

Division

Examples

Powers of indices

Negative exponent

Fractional indices

Summary

Try these!

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