# Simplify expressions

We have to consider certain rules when we operate with exponents. Here follows the most common rules or formulas for operating with exponents or powers:

$a^{b}\cdot a^{c}=a^{(b+c)}$

$(a^{b})^{c}=a^{(b\cdot c)}$

$(ab)^{c}=a^{c}b^{c}$

$(\frac{a}{b})^{c}=\frac{a^{c}}{b^{c}}$

$(\frac{a}{b})^{-c}=\frac{a^{-c}}{b^{-c}}=\frac{b^{c}}{a^{c}}$

$\frac{a^{b}}{a^{c}}=a^{(b-c)}$

$a^{b}=\frac{1}{a^{-b}}$

$a^{0}=1$

Example

Let us study 40.5.
$4^{0.5}\cdot 4^{0.5}=4^{0.5+0.5}=4^{1}=4$
If we multiply 40.5 with itself the answer is 4. Since we know that if we multiply 2 with itself, the answer is also 4. Thus these numbers represent the same thing:
$4^{0.5}\cdot 4^{0.5}=2\cdot 2=4$

$4^{0.5}=4^{1\div 2}=\sqrt{4}=2$

You may know that the more exact term for "the root of" is the "square root of". Sometimes you may choose to emphasize this by writing a two above the root sign:

$\sqrt{4}=\sqrt[2]{4}=2$

For any real numbers a and b the following must be true:

$a^{2}=b,\; a\;is\;the\; square\;root\;of\;b.$

$if\;a^{j}=b\;then\;a\;is\;the\;jth\;root\;of\;b.$

$\sqrt[j]{ab}=\sqrt[j]{a}\cdot \sqrt[j]{b}$

$\sqrt[j]{\frac{a}{b}}=\frac{\sqrt[j]{a}}{\sqrt[j]{b}}$

$b^{\frac{1}{j}}=\sqrt[j]{b}$

j must be a positive integer.

## Video lesson

How can we rewrite 320,2?