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:

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

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








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:


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

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


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



j must be a positive integer.

Video lesson

How can we rewrite 320,2?