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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