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:

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

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

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

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

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

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

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

$$a^{0}=1$$

Example

Let us study 4^0.5.
$$4^{0.5}\cdot 4^{0.5}=4^{0.5+0.5}=4^{1}=4$$
If we multiply 4^0.5 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 32^0,2?