Meaning before procedure
For a ≠ 0, a⁰ = 1 and a^(−r) = 1/a^r. A negative exponent changes reciprocal position; it does not make the value negative.
- Zero exponent
- A nonzero base raised to zero equals 1.
- Negative exponent
- Signals a reciprocal.
- Positive-exponent form
- A final form with no negative exponents.
- Nonzero restriction
- A base moved to a denominator cannot be zero.
Interactive visual model
See how a negative rational exponent creates the reciprocal of a root–power value.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Identify zero and negative exponents.
- Replace zero powers by 1.
- Move negative powers across the fraction bar.
- Apply rational-root meaning and state restrictions.
Watch the reasoning, not just the answer
Example 1: Rewrite x^(−1/2) with a positive exponent.
Example 2: Evaluate 16^(−3/4).
Example 3: Simplify a^(2/3)/a^(5/3).
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
For x ≠ 0, x⁰ equals what?
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Rewrite x^(−1/2) with a positive exponent.
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Apply the method with support
Evaluate 9^(−1/2).
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Evaluate 8^(−2/3).
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Which statement is true?
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Work without step prompts
Simplify a^(1/2)/a^(3/2).
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Rewrite 1/x^(−3/4).
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Evaluate 16⁰ + 4^(1/2).
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Why must a ≠ 0 in a^(−r)?
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Demonstrate durable understanding
Simplify x^(−2/3)x^(5/3).
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Rewrite (ab)^(−1/2).
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Evaluate 25^(−1).
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Simplify y⁰/y^(−1/3).
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Which final form uses only positive exponents?
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Explain, compare, and revise
Rewrite (x^(−1/2)y⁰)/(z^(−2/3)) using only positive exponents.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
Use y⁰ = 1. Move x^(−1/2) to the denominator and z^(−2/3), already in the denominator, to the numerator: z^(2/3)/x^(1/2). The original expression requires nonzero bases where reciprocals occur.
The learner record reveals exact answers and model reasoning only after an attempt.