Rational Exponents
A free Algebra II lesson from the “Rational Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.
Rational exponents connect powers and roots. An exponent of 1/2 means square root, 1/3 means cube root, and m/n means take the nth root and raise to the m power.
What you'll learn
- Rewrite radicals as rational exponents
- Evaluate simple rational exponents
- Use exponent rules
Why it matters: Power-spectrum frequency math, growth-rate models, and engineering yield calculations all use fractional exponents. The form x^(m/n) is the same as the n-th root of x^m, which is the bridge between exponent notation and radical notation.
Worked example
Problem. Evaluate 16^(1/2).
- An exponent of 1/2 means square root.
- sqrt(16) = 4.
- So 16^(1/2) = 4.
Answer: 4
Practice problems
1. Evaluate 25^(1/2).
Show solution
- Warm-up: First identify exactly what the question is asking: Evaluate 25^(1/2).
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Square root of 25 is 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
2. Evaluate 27^(1/3).
Show solution
- Warm-up: First identify exactly what the question is asking: Evaluate 27^(1/3).
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Cube root of 27 is 3.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
Answer: 3
3. Evaluate 8^(2/3).
Show solution
- Core Practice: First identify exactly what the question is asking: Evaluate 8^(2/3).
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Cube root of 8 is 2.
- 2^2 = 4.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
4. Rewrite sqrt(x) as x to what exponent?
Show solution
- Core Practice: First identify exactly what the question is asking: Rewrite sqrt(x) as x to what exponent?
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- Square root means exponent 1/2.
- Check the result by substituting or estimating: the response should match 1/2 and make sense in the original problem.
Answer: 1/2
5. 64^(1/3) equals...
Choices: 4 · 8 · 16 · 32
Show solution
- Challenge: First identify exactly what the question is asking: 64^(1/3) equals...
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- 4^3 = 64.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 4
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