## Basic Exponent Laws I

A power consists of a base an exponent. For the power: $$x^5$$ the base is $x$ and the exponent is $5$.

Exponents are a shorter way to show repeated multiplication of the base. In expanded form, the power $x^5$, would be written as: $$x\times x \times x \times x \times x$$

If you don’t see an exponent on a number or variable, the exponent is $1$.

#### Law of Zero Exponents

You’ll simply need to memorize that for any base, if the exponent is $0$, the answer is $1$: $$x^0=1$$

### Anything to exponent zero is 1.

#### Law of Multiplication

Multiply two powers with the same base: $x^5$ and $x^2$: $$(x^5)(x^2)$$

If you write this in expanded form: $$(x \times x \times x \times x \times x)(x \times x)$$

This would be equivalent to writing: $$(x \times x \times x \times x \times x \times x \times x)=x^7$$

We can conclude that: $$(x^5)(x^2)=x^{5+2}=x^7$$

This will always be true when multiplying powers with the same base. It is called the Law of Multiplication:

### When multiplying powers with the same base, keep the base and add the exponents.

#### Law of Division

Divide two powers with the same base: $x^5$ and $x^2$: $$\frac{x^5}{x^2}$$

If you write this in expanded form: $$\frac{x \times x \times x \times x \times x}{x \times x}$$

Two factors would cancel from the numerator and denominator: $$\frac{x \times x \times x \times x \times x}{x \times x}=x \times x \times x=x^3$$

We can conclude that: $$\frac{x^5}{x^2}=x^{5-2}=x^3$$

This will always be true when dividing powers with the same base. It is called the Law of Division:

### When dividing powers with the same base, keep the base and subtract the exponents.

#### Law of Powers

Raise a power to an even higher exponent. For example, raise $x^5$ to the exponent $2$: $$(x^5)^2$$

If you write this in expanded form: $$(x \times x \times x \times x \times x)(x \times x \times x \times x \times x)$$

This would be equivalent to writing: $$x \times x \times x \times x \times x \times x \times x \times x \times x \times x=x^{10}$$

We can conclude that: $$(x^5)^2=x^{5 \times 2}=x^{10}$$

This will always be true when raising powers to a higher exponent. It is called the Law of Powers:

### Practice

EXAMPLE 1

Simplify then evaluate: $2^{11} \div 2^8$

###### SOLUTION

Using the Law of Division: $$2^{11} \div 2^8=2^{11-8}=2^3$$

Evaluating: $2^3=8$

EXAMPLE 2

Simplify: $(m^0)^{10}$

###### SOLUTION

Using the Law of Powers: $$(m^0)^{10}=m^0$$

We should more properly use the Law of Zero Exponents to write: $m^0=1$

EXAMPLE 3

Find the value of $x$ that makes the equation true: $4^x \times 4^4=4^7$

###### SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: $$x+4=7$$

Therefore: $x=3$