Exponents and Radical Foundations
A free College Algebra lesson from the “Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.
Exponent rules organize repeated multiplication. When multiplying same bases, add exponents; when raising a power to a power, multiply exponents.
What you'll learn
- Use exponent rules
- Simplify powers
- Connect roots and powers
Why it matters: Scaling laws, scientific formulas, compound units, and repeated growth all use exponents to compress multiplication. The rules are not shortcuts to memorize; they describe how repeated factors are grouped.
Worked example
Problem. Simplify (x^3)^4.
- Use the power-to-power rule.
- Multiply 3 and 4.
- The result is x^12.
Answer: x^12
Practice problems
1. Simplify x^4 times x^5.
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify x^4 times x^5.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Add exponents.
- Check the result by substituting or estimating: the response should match x^9 and make sense in the original problem.
Answer: x^9
2. Simplify (a^2)^6.
Show solution
- Core Practice: First identify exactly what the question is asking: Simplify (a^2)^6.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Multiply exponents.
- Check the result by substituting or estimating: the response should match a^12 and make sense in the original problem.
Answer: a^12
3. Simplify b^0 for b not equal to 0.
Show solution
- Challenge: First identify exactly what the question is asking: Simplify b^0 for b not equal to 0.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Any nonzero base to the zero power is 1.
- Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.
Answer: 1
4. Simplify x^5 * x^2.
Choices: x^7 · x^10 · x^3 · 2x^7
Show solution
- Product Rule: First identify exactly what the question is asking: Simplify x^5 * x^2.
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Add exponents with the same base.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x^7
5. Simplify (a^3)^4.
Choices: a^12 · a^7 · a^1 · 4a^3
Show solution
- Power Rule: First identify exactly what the question is asking: Simplify (a^3)^4.
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Multiply the exponents.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: a^12
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