# INTRODUCTION TO INDICES (LAW OF INDICES)

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Indices are simple ways of writing numbers.

To illustrate, in the index form
$3^8$.

The small $8$ is called the index or exponent or power while $3$ is the base.

The index tells you how many of the base numbers to multiply. Hence, if we have $b^4$, it simply means
$b^4=b\times b\times b\times b$

As you would learn subsequently in this blog post, an index can be positive and negative.

LAW OF INDICES

First law
$b^x\times b^y=b^{x+y}$

Example
1. $5^2\times5^3=5^{2+3}=5^5=3125$
2. $x^2\times x^4=x^{2+4}=x^6$

Second law
$b^x÷b^y=b^{x-y}$
This can be also be rewritten as $\frac{b^x}{b^y}=b^{x-y}$

Example
1. $5^3÷5^2=5^{3-2}=5^1=5$
2. $x^4÷x^2=x^{4-2}=x^2$

Third law
$(b^x)^y=b^{x\times y}=b^{xy}$

This can also be interpreted as
$(b^xa^y)^n=b^{xn}a^{yn}$

Example
1. ${(5^2)}^3=5^6=15625$
2. ${(2^4\times3^2)}^3=2^{4\times3}\times3^{2\times3}=2^{12}\times3^6$

Fourth law
$b^0=1$

Example
1. $10^0=1$

Fifth law
$b^{-x}=\frac{1}{b^x}$

Example
1. $7^{-2}=\frac{1}{7^2}=\frac{1}{49}$
2. $5^{-2}=\frac{1}{5^2}=\frac{1}{25}$

Sixth law
$b^{\frac{x}{y}}=(\sqrt[y]{y})^x$

Example
$27^{\frac{2}{3}}=(\sqrt[3]{27})^2=(3)^2=9$

It is important to note that the law of indices can only be used if the numbers are written with the same base. If the numbers have different bases, then, you must solve for the value of each number.

Example 1
Simplify $9x^{-3}\times2x^{8}$

First, we separate the numbers like this:
$9\times2\times x^{-3}\times x^{8}$
$18\times x^{-3+8}$
$18\times x^{5}$
$18x^5$

Related post
Example 2
Simplify $\frac{8x^2y^2}{5x^2y}\times\frac{15x^3y}{16y^3x}$

This translate to
$\frac{8\times15}{5\times16}\times\frac{x^{2+3}}{x^{2+1}}\times\frac{y^{2+1}}{y^{1+3}}$
$\frac{120}{80}\times\frac{x^{5}}{x^{3}}\times\frac{y^{3}}{y^{4}}$

By law of indices
$\frac{3}{2}\times x^{5-3}\times y^{3-4}$
$\frac{3}{2}\times x^{2}\times y^{-1}$
$\frac{3}{2}\times\frac{x^2}{1}\times\frac{1}{y}$
$\frac{3x^2}{2^y}$.

That's all for now. In my next post, you will see some examples that will solidify your understanding of indices.

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