Meaning before procedure
For the same nonzero base, multiply by adding exponents and divide by subtracting exponents: a^r·a^s = a^(r+s) and a^r/a^s = a^(r−s). Rational exponents follow the same rules as integer exponents.
- Same base
- The repeated factor that makes addition/subtraction of exponents valid.
- Product rule
- Add exponents when multiplying like bases.
- Quotient rule
- Subtract denominator exponent from numerator exponent.
- Common denominator
- A shared denominator used to add or subtract fractions.
Interactive visual model
Use fraction controls to rehearse the arithmetic inside exponent rules.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Confirm the bases match.
- Choose add for multiplication or subtract for division.
- Compute the rational-number operation.
- Simplify the exponent and convert form only if useful.
Watch the reasoning, not just the answer
Example 1: Simplify x^(1/2)·x^(1/3).
Example 2: Simplify y^(5/4)/y^(1/2).
Example 3: Simplify a^(−1/3)·a^(4/3).
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
Simplify x^(1/2) · x^(1/3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(3/4) / x^(1/2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Apply the method with support
Simplify x^(2/3) · x^(5/6).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(7/8) / x^(1/4).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(5/6) · x^(1/3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Work without step prompts
Simplify x^(4/5) / x^(1/10).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(3/2) · x^(2/3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(11/12) / x^(5/12).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(2/7) · x^(3/7).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Demonstrate durable understanding
Simplify x^(5/4) / x^(3/8).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(1/6) · x^(5/6).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(9/10) / x^(2/5).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(7/3) · x^(1/6).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify x^(13/8) / x^(5/8).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Explain, compare, and revise
Simplify z^(7/6)·z^(1/3)/z^(1/2) and explain each exponent operation.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
Add for the product and subtract for the quotient: 7/6 + 1/3 − 1/2 = 7/6 + 2/6 − 3/6 = 6/6 = 1. The expression simplifies to z.
The learner record reveals exact answers and model reasoning only after an attempt.