Exponent Rules and Rational Exponents

5.4: Exponent Rules and Rational Exponents. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 4 of 63 parts Practice Flashcards

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Products and quotients of powers

Objective: Combine equal-base powers by exponent laws.

Prerequisite skills: integer exponents and fraction arithmetic. Multiplying equal-base powers adds exponents. Dividing subtracts exponents when the base is nonzero. A power of a power multiplies exponents. These laws concern products and quotients, not sums. Common error: multiplying exponents in an ordinary product of powers. Restrictions and interpretation: For real rational powers, even-denominator roots require nonnegative radicands; a negative power excludes zero.

Formulas

$a^m a^n = a^(m+n); a^m/a^n = a^(m−n), a ≠ 0$

Worked examples

Try itFor nonzero a, a^25/a^22 = a^k. What is k?Show answer

Subtract denominator exponent from numerator exponent: k = (25) − (22) = 3. The nonzero-base condition makes division valid. Therefore the requested answer is 3.

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Products and quotients of powers

Objective: Combine equal-base powers by exponent laws.

Formulas

$a^m a^n = a^(m+n); a^m/a^n = a^(m−n), a ≠ 0$

Worked examples

Try itFor nonzero a, a^25/a^22 = a^k. What is k?Show answer

Subtract denominator exponent from numerator exponent: k = (25) − (22) = 3. The nonzero-base condition makes division valid. Therefore the requested answer is 3.

Rational powers

Objective: Interpret a fractional exponent as a root followed by a power.

For a nonnegative base, a^(m/n) = (nth root of a)^m. When n is even, restrict to real roots of nonnegative numbers. Use known perfect powers to avoid approximate calculation. Common error: dividing the base by the exponent denominator. Restrictions and interpretation: For real rational powers, even-denominator roots require nonnegative radicands; a negative power excludes zero.

Formulas

$a^(m/n) = (ⁿ√a)^m$

Worked examples

Try itEvaluate 576^(3/2).Show answer

Take the square root of 576, which is 24, then cube it: 24³ = 13824. The positive base meets the real-domain requirement. Therefore the requested answer is 13824.

Equivalent exponential powers

Objective: Use a common base to compare structured exponents.

Rewrite each base as a power of the same positive number. Apply the power-of-a-power rule, then combine exponents. This often converts a complicated expression into one integer exponent. Common error: adding exponents when raising a power to another power. Restrictions and interpretation: For real rational powers, even-denominator roots require nonnegative radicands; a negative power excludes zero. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$(a^p)^q = a^(pq)$

Worked examples

Try itIf (4^24)(8²) = 2^k, what is k?Show answer

Since 4 = 2² and 8 = 2³, the expression is 2^48 × 2⁶. Add exponents: k = 54. Therefore the requested answer is 54.

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