Exponent Rules
A free Algebra I lesson from the “Exponents and Polynomials” unit, with a worked example and practice problems including step-by-step solutions.
Exponent rules keep repeated multiplication organized. The operation changes the exponent: multiply powers by adding exponents, divide by subtracting, and raise a power to a power by multiplying exponents.
What you'll learn
- Multiply and divide powers with the same base
- Raise powers to powers
- Use zero and negative exponents
Why it matters: Exponent rules keep scientific notation, computer storage sizes, population models, and repeated-growth calculations manageable.
Worked example
Problem. Simplify (x^3)^4.
- A power raised to a power means repeated groups.
- Multiply the exponents: 3 x 4 = 12.
- (x^3)^4 = x^12.
Answer: x^12
Practice problems
1. Simplify x^4 times x^2.
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify x^4 times x^2.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Same base multiplication means add exponents.
- 4 + 2 = 6.
- Check the result by substituting or estimating: the response should match x^6 and make sense in the original problem.
Answer: x^6
2. Simplify y^7 divided by y^3.
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify y^7 divided by y^3.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Same base division means subtract exponents.
- 7 - 3 = 4.
- Check the result by substituting or estimating: the response should match y^4 and make sense in the original problem.
Answer: y^4
3. Simplify (a^2)^3.
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify (a^2)^3.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Power to a power means multiply exponents.
- 2 x 3 = 6.
- Check the result by substituting or estimating: the response should match a^6 and make sense in the original problem.
Answer: a^6
4. Simplify m^5 times m^6.
Show solution
- Core Practice: First identify exactly what the question is asking: Simplify m^5 times m^6.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Add exponents with the same base.
- 5 + 6 = 11.
- Check the result by substituting or estimating: the response should match m^11 and make sense in the original problem.
Answer: m^11
5. Simplify b^9 divided by b^4.
Show solution
- Core Practice: First identify exactly what the question is asking: Simplify b^9 divided by b^4.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Subtract exponents.
- 9 - 4 = 5.
- Check the result by substituting or estimating: the response should match b^5 and make sense in the original problem.
Answer: b^5
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