g10-06-02 · Core topics
A fractional exponent names a root
What does 16^(3/4) ask you to do?
1 MIN · PREDICT ON PAPER
Take a fourth root and cube it; predict whether reversing those two operations changes the result.
Discuss or write first. No typed answer is required to continue.
The positive 4th root is 2; raising it to 3 gives 8.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
For positive a, a^(p/q)=(q√a)ᵖ with integer p and positive integer q. Here 16^(3/4)=2³=8. Negative p then takes a reciprocal.
Conditions: Use a>0 so all rational exponents have one consistent real definition. This activity rejects a≤0 rather than extending selected odd-root cases.
Mathematical justification: The positive qth root is the unique positive r with r^q=a. Define a^(p/q)=rᵖ; reducing p/q preserves the same positive value.
1 MIN · EXIT CHECK
Evaluate 27^(2/3).
Check after answering
9, since the cube root of 27 is 3.
Teacher guidance and extension
Before: Real-number operations and reading coordinates; write a prediction on paper.
Ask: Label numerator “power” and denominator “root”; keep the positive-base condition beside the rule.
Watch for: a^(p/q)=(aᵖ)/q.
Challenge: Explain why 16^(−3/4)=1/8.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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