Learning Objectives
Apply exponent rules with the same base and state nonzero conditions.
Interpret zero and negative exponents.
Simplify square roots and distinguish a principal square root from equation solutions.
Prerequisites: multiplication, fractions, signed numbers, and algebraic substitution. Work over the real numbers unless a question explicitly introduces complex numbers.
1. Exponents Count Repeated Factors
For a positive integer n, aⁿ is a product of n copies of a. This explains the product rule aᵐaⁿ = aᵐ⁺ⁿ: multiplying joins the repeated factors. The power rule (aᵐ)ⁿ = aᵐⁿ counts m factors in each of n groups.
Rule for integer exponents | Condition |
|---|---|
aᵐaⁿ = aᵐ⁺ⁿ | Expressions must be defined |
aᵐ/aⁿ = aᵐ⁻ⁿ | a ≠ 0 |
(aᵐ)ⁿ = aᵐⁿ | Expressions must be defined |
a⁰ = 1 | a ≠ 0 |
a⁻ⁿ = 1/aⁿ | a ≠ 0 |
Worked Example: Simplify a Quotient
Simplify (6x⁵y²)/(3x²y⁴), with x and y nonzero. Divide coefficients and subtract exponents for matching bases.
The result is 2x³y⁻² = 2x³/y².
The original denominator excludes x = 0 as well as y = 0. Even though the simplified formula has no x in its denominator, the original restriction remains when claiming equivalence.
2. A Negative Exponent Means a Reciprocal
A negative exponent does not make the value negative. For example, 2⁻³ = 1/8, while (−2)³ = −8. A negative base and a negative exponent describe different features of an expression.
Example: Parentheses Matter
(−3)² = 9 because the base is −3. In contrast, −3² = −9 because the exponent applies to 3 before the outside negative sign.
Also, 3⁻² = 1/9. Keep the base, exponent, and any outside sign distinct.
3. Simplify Square Roots by Extracting Squares
For nonnegative a and b, √(ab) = √a√b. To simplify √72, factor 72 as 36 × 2, giving 6√2. This product rule does not allow splitting a sum: √(a + b) is generally not √a + √b.
Worked Example: Combine Like Radicals
Simplify √50 + 3√8. Since √50 = 5√2 and √8 = 2√2, the expression becomes 5√2 + 6√2.
The result is 11√2. The radical parts match, so their coefficients can be added.
By contrast, √2 + √3 does not combine into √5.
4. Roots and Equations Are Different
The symbol √25 denotes the principal, nonnegative square root, 5. The equation x² = 25 has two real solutions, x = 5 and x = −5. Similarly, √(x²) = |x|, not always x, because a principal square root is never negative.
For a > 0, a^(1/n) denotes the positive nth root and a^(m/n) can be interpreted using roots and powers. Positive bases avoid ambiguity when manipulating general rational exponents. Check any additional domain conditions before extending a rule to negative bases.
Common Mistakes
Exponent product rules apply to multiplication, not addition: x² + x³ is not x⁵.
Do not split a square root across a sum.
Keep original domain exclusions after simplification.
5. Practice
1. Simplify x³x⁴/x² for x ≠ 0.
2. Evaluate 4⁻² and (−4)².
3. Simplify √108 − √12.
4. Evaluate √(x²) when x = −7.
Worked Solutions
1. Add and subtract exponents: x^(3 + 4 − 2) = x⁵, retaining x ≠ 0.
2. 4⁻² = 1/16, while (−4)² = 16.
3. √108 = 6√3 and √12 = 2√3, so the difference is 4√3.
4. √(x²) = |x| = 7. The principal square root is nonnegative.
6. Summary
- Use exponent rules only for the operations and domains they describe.
- Negative exponents create reciprocals, not negative signs.
- Extract perfect-square factors before combining radicals.
- Distinguish a principal root from the solutions of a power equation.