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:

When raising a power to a higher exponent, keep the base and multiply the exponents.

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

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