Powers and exponents

We know how to calculate the expression 5 x 5. This expression can be written in a shorter way using something called exponents.

$\\ 5\cdot 5=5^{2} \\$

An expression that represents repeated multiplication of the same factor is called a power.

The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.

3¹	3 to the first power	3
4²	4 to the second power or 4 squared	4 ∙ 4
5³	5 to the third power or 5 cubed	5 ∙ 5 ∙ 5
2⁶	2 to the power of six	2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2

Example:

Write these multiplications like exponents

$\\ 5\cdot 5\cdot 5=5^{3}\\ 4\cdot 4\cdot 4\cdot 4\cdot 4=4^{5} \\ 3\cdot 3\cdot 3\cdot 3=3^{4} \\$

Multiplication: If two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents.

The rule:

$\\x^{a}\cdot x^{b}=x^{a+b}\\$

Example:

$\\4^{2}\cdot 4^{5}=\left ( 4\cdot 4 \right )\cdot \left ( 4\cdot 4\cdot 4\cdot 4\cdot 4 \right )=4^{7}=4^{2+5}\\$

Division: If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents.

The rule:

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

Example:

$\\ \frac{4^{2}}{ 4^{5}}=\frac{{\color{red} \not}{4}\cdot {\color{red} \not}{4}}{{\color{red} \not}{4}\cdot {\color{red} \not}{4}\cdot 4\cdot 4\cdot 4}=\frac{1}{4^{3}}=4^{-3}=4^{2-5} \\$