Laws of Exponents
Apply the rules of integer exponents to write equivalent expressions: multiply powers by adding indices, divide by subtracting, raise a power to a power by multiplying, and interpret zero and negative exponents (3² × 3⁻⁵ = 3⁻³ = 1/27)
What a learner can do afterwards
- Simplify products and quotients of powers with the same base (e.g., 5³ × 5⁴ = 5⁷, 2⁸ ÷ 2³ = 2⁵)
- Evaluate expressions with zero and negative exponents (e.g., 7⁰ = 1, 2⁻³ = 1/8)
- Spot and correct a common exponent error, such as claiming 3² × 3³ = 9⁵
The lesson
An exponent tells you how many times to multiply a number by itself. When two powers share the same base, like 2³ and 2⁴, you can multiply them by adding the exponents: 2³ × 2⁴ = 2⁷. Here's why: 2³ is three 2s multiplied together, and 2⁴ is four more 2s. Put them together and you get seven 2s multiplied together, which is 2⁷.
Farah simplifies 5⁶ ÷ 5². She keeps the base 5 and subtracts the exponents: 6 − 2 = 4. So 5⁶ ÷ 5² = 5⁴. Dividing cancels matching 5s from the top and bottom, leaving four 5s multiplied together.
There's also a rule for a power raised to another power, like (2³)². This means 2³ multiplied by itself twice, so you multiply the exponents instead of adding them: 3 × 2 = 6. That gives (2³)² = 2⁶.
Zero and negative exponents follow the pattern from dividing. Since 2³ ÷ 2³ = 2⁰, and a number divided by itself is always 1, you get 2⁰ = 1. Go one step further, like 2² ÷ 2⁵ = 2⁻³, and a negative exponent means flip it into a fraction: 2⁻³ = 1/2³ = 1/8. Watch out for a common mistake: 3² × 3³ is 3⁵, not 9⁵. The base stays 3; only the exponents add.
Same base: add exponents to multiply, subtract to divide, multiply for a power of a power, and remember zero gives 1 while negative flips to a fraction.
Watch it
Where it sits
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Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.