Laws of Exponents

Are you having issues applying integral powers in mathematical expressions? In this lesson, we will discuss and apply the laws for using exponents. Before starting the application process let’s first discuss the details for each law.

Product Rule:     a^{n .} a^m = a^{n +m}   

When multiplying two numbers that have the same base the exponents are added.

Quotient Rule: aⁿ/a^m = a ^{n – m} 

When dividing two numbers that are the same base the exponents are subtracted.

Power of a Product Rule:  (ab)^x  = a^xb^x

When a term raised to a power is equal to the product of its factors raised to the same power. 

Zero Product: a⁰ = 1

Any number (excluding 0) raised to the power of 0 is always equal to 1. 

Negative Product: a^-x = 1/aⁿ 

A value raised to a negative power can be written as a fraction with the positive power of that number in the denominator.

Examples

1. 2² • 2³ = 2 ^{2 + 3} = 2⁵
2. x² • x⁵ = x²⁺⁵ = x⁷ 
3. a⁷• a  = a^{7 + 1} = a⁸
4. (4 • 5)² = 4² • 5² = 16 • 25 = 400
5. ( x • y)³  = x³ y³
6. Rewrite : 2^-3 =1/2³
7. Rewrite: x^-4 = 1/x⁴
8. 2⁰ = 1
9. 4⁷/ 4³ = 4^7-3 = 4⁴

Examples 1-3 use the product rule. Examples 4-5 use the power of a product rule and examples 6-7 use the negative product rule. Example 8 uses the zero product rule while example 9 uses the quotient rule.

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