Meaning before procedure
A power of a power multiplies exponents: (a^r)^s = a^(rs). A power distributes to every factor in a product and to numerator and denominator factors in a quotient when defined.
- Power of a power
- Multiply the inner and outer exponents.
- Power of a product
- Apply the exponent to every factor.
- Power of a quotient
- Apply the exponent to numerator and denominator.
- Grouping
- Parentheses show the full expression affected by the outer power.
Interactive visual model
Compare a nested power with the product of its exponents.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Identify the complete powered group.
- Distribute the outer exponent to every factor if needed.
- Multiply nested exponents.
- Simplify coefficients and rational exponents.
Watch the reasoning, not just the answer
Example 1: Simplify (x^(2/3))^(3/4).
Example 2: Simplify (a²b³)^(1/2).
Example 3: Simplify ((m^4)/(n²))^(1/2).
Predict, decide, check, and revise
Build a valid reasoning path
Choose the decisions in the order a careful solver would use them for this concept.
The reasoning path changes as each decision is selected.
Check the foundation
Simplify (x^(1/2))^(3/2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify (x^(2/3))^(3/4).
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Apply the method with support
Simplify (x^(3/4))^(2/3).
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Simplify (x^(5/6))^(3/5).
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Simplify (x^(7/8))^(4/7).
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Work without step prompts
Simplify (x^(2/5))^(5/3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify (x^(3/2))^(2/5).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify (x^(4/3))^(3/8).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify (x^(5/4))^(2/5).
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Demonstrate durable understanding
Simplify (x^(1/3))^(9/2).
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Simplify (x^(7/6))^(3/7).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify (x^(5/8))^(4/5).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify (x^(9/10))^(5/6).
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Simplify (x^(11/12))^(6/11).
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Explain, compare, and revise
Simplify [(x^(3/2)y^(1/2))]² and justify why both factors receive the outer power.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
The parentheses make the entire product the base of the outer square. Multiply each exponent by 2: x^(3/2·2)y^(1/2·2) = x³y.
The learner record reveals exact answers and model reasoning only after an attempt.