LEARN · EXPLAIN · REVISE

Simplify for real x, including negative x.

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

TOPIC 01

Exponents

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

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

01 / Standard#Your turn

\sqrt{x^2}

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

|x|.

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

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

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

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.