Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Exponents

Read the idea, work independently, then explain what changed.

高一必修 第一册(A版).pdf · 4.1 · PDF 111 / printed page 104

Revisit first: Applications of functions I

TOPIC 01

Exponents

Build understanding of exponents through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Interpret and use exponents with domain checks.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Exponent laws

Use common-base product and quotient laws within their valid domains.

aman=am+na^ma^n=a^{m+n}

Roots

Principal even roots are nonnegative; odd roots allow negative arguments.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in exponents changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Exponential graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Exponential graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Simplify the product.

22⋅232^{2}\cdot2^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
Add exponents when multiplying equal bases.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    22+3=25=322^{2+3}=2^{5}=32

  3. The bases must be the same for this rule.

The requested value is 32.

Checks and common pitfalls: The bases must be the same for this rule.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Evaluate the negative power.

2−22^{-2}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
A negative exponent indicates a reciprocal.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    2−2=1/42^{-2}=1/4

  3. It does not mean the answer is negative.

The requested value is 0.25.

Checks and common pitfalls: It does not mean the answer is negative.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Compare the powers without decimal approximations.

(1/2)2,(1/2)3(1/2)^2,\quad(1/2)^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
The common base is between zero and one.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    (1/2)2−(1/2)3=(2−1)/8>0(1/2)^2-(1/2)^3=(2-1)/8>0

  3. Increasing a positive exponent reduces a power with base between zero and one.

The squared value is larger.

Checks and common pitfalls: Increasing a positive exponent reduces a power with base between zero and one.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Simplify the product.

23⋅232^{3}\cdot2^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
Add exponents when multiplying equal bases.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    23+3=26=642^{3+3}=2^{6}=64

  3. The bases must be the same for this rule.

The requested value is 64.

Checks and common pitfalls: The bases must be the same for this rule.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Evaluate the negative power.

3−23^{-2}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
A negative exponent indicates a reciprocal.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    3−2=1/93^{-2}=1/9

  3. It does not mean the answer is negative.

The requested value is 0.11111111.

Checks and common pitfalls: It does not mean the answer is negative.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Evaluate the rational power.

272/327^{2/3}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
Take the cube root before squaring.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    (27)2/3=(273)2=9(27)^{2/3}=(\sqrt[3]{27})^2=9

  3. The chosen positive base makes both orders of calculation valid.

The requested value is 9.

Checks and common pitfalls: The chosen positive base makes both orders of calculation valid.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Simplify for a>0.

a7a3\frac{a^{7}}{a^{3}}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
Subtract exponents when dividing equal nonzero bases.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    a7−3=a4a^{7-3}=a^4

  3. The nonzero base is required for division.

a⁴.

Checks and common pitfalls: The nonzero base is required for division.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Explain why the rule a⁰=1 must not be used to assign 0⁰ here.

a0=1(a≠0)a^0=1\quad(a\ne0)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
Examine the denominator in the derivation.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    am/am=1=a0(a≠0)a^m/a^m=1=a^0\quad(a\ne0)

  3. The usual real exponent rule excludes a=0 in this derivation.

The rule was derived using division by a nonzero power.

Checks and common pitfalls: The usual real exponent rule excludes a=0 in this derivation.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Simplify for real x, including negative x.

x2\sqrt{x^2}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
The principal square root is nonnegative.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    x2=∣x∣≥0\sqrt{x^2}=|x|\ge0

  3. Replacing it by x would fail when x<0.

|x|.

Checks and common pitfalls: Replacing it by x would fail when x<0.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Compare the powers without decimal approximations.

(1/3)2,(1/3)3(1/3)^2,\quad(1/3)^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
The common base is between zero and one.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    (1/3)2−(1/3)3=(3−1)/27>0(1/3)^2-(1/3)^3=(3-1)/27>0

  3. Increasing a positive exponent reduces a power with base between zero and one.

The squared value is larger.

Checks and common pitfalls: Increasing a positive exponent reduces a power with base between zero and one.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Simplify the product.

24⋅232^{4}\cdot2^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
Add exponents when multiplying equal bases.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    24+3=27=1282^{4+3}=2^{7}=128

  3. The bases must be the same for this rule.

The requested value is 128.

Checks and common pitfalls: The bases must be the same for this rule.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Evaluate the negative power.

4−24^{-2}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
A negative exponent indicates a reciprocal.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    4−2=1/164^{-2}=1/16

  3. It does not mean the answer is negative.

The requested value is 0.0625.

Checks and common pitfalls: It does not mean the answer is negative.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Evaluate the rational power.

642/364^{2/3}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
Take the cube root before squaring.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    (64)2/3=(643)2=16(64)^{2/3}=(\sqrt[3]{64})^2=16

  3. The chosen positive base makes both orders of calculation valid.

The requested value is 16.

Checks and common pitfalls: The chosen positive base makes both orders of calculation valid.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for exponents?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Interpret and use exponents with domain checks.
    • Exponent laws: Use common-base product and quotient laws within their valid domains.
      aman=am+na^ma^n=a^{m+n}
    • Roots: Principal even roots are nonnegative; odd roots allow negative arguments.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Use common-base product and quotient laws within their valid domains.
    • Expected reasoning: Principal even roots are nonnegative; odd roots allow negative arguments.
    • Expected correction: The square root of x² is |x|, not always x.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗