Same base: combine exponents
Multiply powers with the same base by adding exponents; divide by subtracting exponents.
Example and .
Chapter 06 of 10
Exponents
What patterns let us work efficiently with repeated multiplication?
Exponents compress repeated multiplication. The exponent rules are consequences of that meaning—not disconnected tricks. Expanding a small example is often the fastest way to remember which rule applies.
The big ideas
Understand these before you worry about speed.
A power multiplies exponents
When a power is raised to another power, multiply the exponents. Apply an outside exponent to every factor in a product.
Example(2x³)² = 4x⁶.
Zero and negative exponents
A nonzero base to the zero power is 1. A negative exponent indicates a reciprocal.
Example for .
Scientific notation uses powers of ten
Write a nonzero number as a coefficient whose absolute value is at least 1 but less than 10, multiplied by a power of ten. Moving the decimal left creates a positive exponent; moving it right creates a negative exponent. The number of places moved is the exponent's magnitude.
Example because the decimal moves four places to the right.
Worked example
Follow the reason for each step—not just the symbols.
Problem
Simplify . Write with positive exponents.
- Divide coefficients: .
- Subtract x exponents: .
- Subtract y exponents: .
- Rewrite as .
Answer
Try it yourself
Work on paper first. Open each answer only when you are ready to check.
1
Simplify a⁴ · a⁷.
Check answer
a¹¹2
Simplify (3x²)³.
Check answer
27x⁶3
Rewrite 5x⁻² with positive exponents.
Check answer
4
Write 0.00072 in scientific notation.
Check answer
7.2 × 10⁻⁴Practice and tools
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