# 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.

3^{1} |
3 to the first power | 3 |

4^{2} |
4 to the second power or 4 squared | 4 ∙ 4 |

5^{3} |
5 to the third power or 5 cubed | 5 ∙ 5 ∙ 5 |

2^{6} |
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**

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

A negative exponent is the same as the reciprocal of the positive exponent.

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

**Example**

$$2^{-3}=\frac{1}{2^{3}}$$

When you raise a product to a power you raise each factor with a power

$$(x\cdot y)^{a}=x^{a}\cdot y^{a}$$

**Example**

$$(2x)^{4}=2^{4}\cdot x^{4}=16x^{4}$$

The rule for the power of a power and the power of a product can be combined into the following rule:

$$(x^{a}\cdot y^{b})^{z}=x^{a\cdot z}\cdot y^{b\cdot z}$$

**Example**

$$(x^{3}\cdot y^{4})^{2}=x^{3\cdot 2}\cdot y^{4\cdot 2}=x^{6}\cdot y^{8}$$

**Video lessons**

Rewrite the expressions

$$2\cdot 2\cdot 2$$

$$x\cdot x\cdot x\cdot x\cdot x$$

$$3^{4}$$

$$x^{3}$$

Simplify the expression

$$\left ( x^{2}\cdot y^{3}\cdot z^{5} \right )^{3}$$