What you will be able to do
- Add exponents when multiplying like bases.
- Multiply exponents for a power of a power.
- A negative exponent means reciprocal.
- Simplify radicals by extracting perfect-square factors.
Core exponent rules
| Rule | Condition | Example |
|---|---|---|
| aᵐaⁿ = aᵐ⁺ⁿ | Same base, multiplication | x³x⁴ = x⁷ |
| aᵐ/aⁿ = aᵐ⁻ⁿ | Same nonzero base, division | y⁶/y² = y⁴ |
| (aᵐ)ⁿ = aᵐⁿ | Power of a power | (x³)² = x⁶ |
| a⁰ = 1 | a ≠ 0 | 7⁰ = 1 |
| a⁻ⁿ = 1/aⁿ | a ≠ 0 | x⁻² = 1/x² |
Do not apply product rules to addition
Worked example 1
Simplify x² + x³.
- 1.The terms are added, not multiplied.
- 2.Their exponents cannot be combined.
- 3.The expression can be factored as x²(1 + x).
Answer: x² + x³ or x²(1 + x)
Check: x⁵ would be wrong unless the original operation were x² · x³.
Simplify square roots using perfect squares
Worked example 1
Simplify √72.
- 1.Factor 72 as 36 · 2.
- 2.√72 = √36 · √2.
- 3.√36 = 6.
Answer: 6√2
Check: (6√2)² = 36 · 2 = 72.
Connect exponents to scientific notation
Worked example 1
Write 0.00042 in scientific notation.
- 1.Move the decimal four places right to make 4.2.
- 2.Moving right gives a negative exponent.
Answer: 4.2 × 10⁻⁴
Check: 10⁻⁴ = 0.0001; 4.2 times that is 0.00042.
Check the operation before choosing a rule
Learn exponential functionsReview factoring polynomials
- Circle the operation connecting expressions.
- Confirm that bases match for product or quotient rules.
- Keep parentheses around a powered product or quotient.
- Rewrite negative exponents as reciprocals at the end.
- Estimate radical size between nearby perfect squares.
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Useful answers
Frequently asked questions
Why is any nonzero number to the zero power 1?
The quotient rule gives aᵐ/aᵐ = a⁰, while any nonzero quantity divided by itself equals 1.
Is √a + √b equal to √(a + b)?
Not generally. For example, √9 + √16 = 7 while √25 = 5.
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Sources, scope and accuracy
The worked examples and practice questions on this page are original Cambridge Veritas learning materials, not released STAAR items. Curriculum and testing details can change; use the linked Texas Education Agency pages for current official information. Cambridge Veritas is not affiliated with or endorsed by TEA.