g10-06-01 · Core topics
A negative exponent is a reciprocal
Does 2⁻³ mean a negative number or a reciprocal?
1 MIN · PREDICT ON PAPER
Continue the sequence 2³,2²,2¹,2⁰ one step at a time to exponent −3.
Discuss or write first. No typed answer is required to continue.
2^-3 = 0.125. A negative exponent takes a reciprocal.
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
Reducing the exponent by one divides by the nonzero base. Thus a⁰=1 and a⁻ⁿ=1/aⁿ for a≠0. The minus sign in the exponent does not negate the value.
Conditions: The exponent here is an integer. Zero to a negative exponent and 0⁰ are not defined in this task.
Mathematical justification: For a≠0, preserving aⁿa⁻ⁿ=a⁰=1 forces a⁻ⁿ=1/aⁿ. This algebra, rather than a few sequence terms, establishes the identity.
1 MIN · EXIT CHECK
Find 5⁻².
Check after answering
1/25.
Teacher guidance and extension
Before: Real-number operations and reading coordinates; write a prediction on paper.
Ask: Ask students to name the repeated operation when the exponent decreases.
Watch for: a⁻ⁿ=−aⁿ.
Challenge: For a=−2, compare odd and even negative exponents.
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.
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