PAPER · REASONING · PRACTICE

Square roots: meaning, simplification and domain

Distinguish a square root from the solutions of a squared equation, and retain every real-domain condition.

40 minutes · Year 8, lower volume · text pp. 1–4 · textbook lessons 1–2

Textbook reference: Year 8, lower volume · Chapter 16: Square-root expressions · lessons 1–2

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

3 min: Predict the values of $\sqrt{x^{2}}$ for x=3, −3 and 0. Does the original sign survive? Write a reason before calculating. 8 min: On paper, make columns for x, its square and the principal root of that square. Then test the expression √(x−3) at x=1,3,7; mark which inputs are allowed over the reals. 5 min: Explain $\sqrt{x^{2}}=|x|$ and the condition $x-3\geq0$ . A few examples suggest the rule; justify it using the definition of the nonnegative root. 2 min: Individually simplify $\sqrt{(-5)^{2}}$ and distinguish it from solving $x^{2}=25$ .

Organise the key idea

The value of $\sqrt{a}$ is nonnegative, and $\sqrt{x^{2}}=|x|$ .

Open the original chapter

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

Simplify exactly. State any restriction that is needed.

\sqrt{(-7)^{2}}

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $7$ and explain the simplification.

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
An intermediate form is $\sqrt{49}=7$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

Simplify exactly. State any restriction that is needed.

\sqrt{50}

Complete the square-factor split: $50=25\times\square$ .
Then $\sqrt{50}=\sqrt{25}\sqrt{\square}=\square\sqrt{2}$ .
Check that the coefficient is nonnegative and the square of the result is 50.

Look for the largest perfect-square factor of 50; try 25.
$50=25\times2$ ; split the square root of this product.
Evaluate the square root of 25 and retain the factor √2; check by squaring.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\sqrt{25\times2}=5\sqrt{2}$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

Simplify exactly. State any restriction that is needed.

\sqrt{49}

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $7$ and explain the simplification.

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
The simplified form is $7$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

Is this expression defined over the real numbers? Explain.

\sqrt{-4}

Start with the original radicands.
Zero is allowed under a square-root sign.
Check every endpoint in the original expression.

Every radicand must be nonnegative.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $\varnothing$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Find all real values of x for which the expression is defined.

\sqrt{x-3}

Start with the original radicands.
Zero is allowed under a square-root sign.
Check every endpoint in the original expression.

Every radicand must be nonnegative.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $x\geq3$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

Find all real values of x for which the expression is defined.

\sqrt{5-2x}

Start with the original radicands.
Zero is allowed under a square-root sign.
Check every endpoint in the original expression.

Every radicand must be nonnegative.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $x\leq\frac{5}{2}$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

Simplify exactly. State any restriction that is needed.

\sqrt{x^{2}}

A square root cannot be negative, even when x is negative.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $|x|$ and explain the simplification.

A square root cannot be negative, even when x is negative.
The simplified form is $|x|$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

Find all real values of x for which the expression is defined.

\frac{1}{\sqrt{x+2}}

Start with the original radicands.
Do not allow a zero denominator.
Check every endpoint in the original expression.

Every radicand must be nonnegative; a square root in a denominator must be strictly positive.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $x>-2$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Simplify exactly. State any restriction that is needed.

\sqrt{0}

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $0$ and explain the simplification.

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
The simplified form is $0$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

A learner writes $\sqrt{25}=\pm5$ . Give the correct value and explain the error.

\sqrt{25}

The equation x²=25 has two solutions; the square-root symbol itself denotes only 5.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $5$ and explain the simplification.

The equation x²=25 has two solutions; the square-root symbol itself denotes only 5.
The simplified form is $5$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

Simplify exactly. State any restriction that is needed.

\sqrt{\frac{1}{4}}

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $\frac{1}{2}$ and explain the simplification.

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
The simplified form is $\frac{1}{2}$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

Find all real values of x for which the expression is defined.

\sqrt{x-1}+\sqrt{4-x}

Start with the original radicands.
Zero is allowed under a square-root sign.
Check every endpoint in the original expression.

Every radicand must be nonnegative.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $1\leq x\leq4$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

Simplify exactly. State any restriction that is needed.

\sqrt{(0-2)^{2}}

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $2$ and explain the simplification.

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
An intermediate form is $|0-2|=2$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Simplify exactly. State any restriction that is needed.

(\sqrt{11})^{2}

Squaring the nonnegative square root restores its nonnegative radicand.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $11$ and explain the simplification.

Squaring the nonnegative square root restores its nonnegative radicand.
The simplified form is $11$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

Find all real values of x for which the expression is defined.

\frac{\sqrt{2x+1}}{\sqrt{3-x}}

Start with the original radicands.
Do not allow a zero denominator.
Check every endpoint in the original expression.

Every radicand must be nonnegative; a square root in a denominator must be strictly positive.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $-\frac{1}{2}\leq x<3$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

Use this value to disprove the claim $\sqrt{x^{2}}=x$ for every real x.

\sqrt{(-2)^{2}}

At x=−2, the root is 2 while x is −2. The correct identity uses absolute value.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $2$ and explain the simplification.

At x=−2, the root is 2 while x is −2. The correct identity uses absolute value.
The simplified form is $2$ .
Check the original expression and the sign of the square-root value.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Preparation and assessment notes

Prerequisites

Squares, nonnegative numbers and linear inequalities.

Look for in students’ work

Use the two prerequisite checks to choose support. Project the worked and guided tasks, then collect the eight paper core questions. Follow-ups and extensions are optional; accept other complete valid methods. This page does not save learners’ paper answers.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original chapter-core practice after the cited lower-volume textbook lessons. References identify skills; textbook exercises are not reproduced.

大湾区初中数学三维课堂（八年级下册）正文.pdf

Year 8, lower volume · text pp. 1–4 · textbook lessons 1–2

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Square roots: meaning, simplification and domain

Year 8, lower volume · text pp. 1–4 · textbook lessons 1–2 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Simplify exactly. State any restriction that is needed.

\sqrt{(-7)^{2}}

Guided attempt2

Simplify exactly. State any restriction that is needed.

\sqrt{50}

Complete the square-factor split: $50=25\times\square$ .
Then $\sqrt{50}=\sqrt{25}\sqrt{\square}=\square\sqrt{2}$ .
Check that the coefficient is nonnegative and the square of the result is 50.

Prerequisite3

Simplify exactly. State any restriction that is needed.

\sqrt{49}

Prerequisite4

Is this expression defined over the real numbers? Explain.

\sqrt{-4}

Square roots: meaning, simplification and domain

Year 8, lower volume · text pp. 1–4 · textbook lessons 1–2 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Find all real values of x for which the expression is defined.

\sqrt{x-3}

Independent practice6

Find all real values of x for which the expression is defined.

\sqrt{5-2x}

Independent practice7

Simplify exactly. State any restriction that is needed.

\sqrt{x^{2}}

Independent practice8

Find all real values of x for which the expression is defined.

\frac{1}{\sqrt{x+2}}

Exit question9

Simplify exactly. State any restriction that is needed.

\sqrt{0}

Exit question10

A learner writes $\sqrt{25}=\pm5$ . Give the correct value and explain the error.

\sqrt{25}

Square roots: meaning, simplification and domain — Further practice

Year 8, lower volume · text pp. 1–4 · textbook lessons 1–2

Further practice11

Simplify exactly. State any restriction that is needed.

\sqrt{\frac{1}{4}}

Further practice12

Find all real values of x for which the expression is defined.

\sqrt{x-1}+\sqrt{4-x}

Further practice13

Simplify exactly. State any restriction that is needed.

\sqrt{(0-2)^{2}}

Further practice14

Simplify exactly. State any restriction that is needed.

(\sqrt{11})^{2}

Extension15

Find all real values of x for which the expression is defined.

\frac{\sqrt{2x+1}}{\sqrt{3-x}}

Extension16

Use this value to disprove the claim $\sqrt{x^{2}}=x$ for every real x.

\sqrt{(-2)^{2}}

Square roots: meaning, simplification and domain — Teacher answers

Year 8, lower volume · text pp. 1–4 · textbook lessons 1–2

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Squares, nonnegative numbers and linear inequalities.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

3′ Predict the values of $\sqrt{x^{2}}$ for x=3, −3 and 0. Does the original sign survive? Write a reason before calculating.
8′ On paper, make columns for x, its square and the principal root of that square. Then test the expression √(x−3) at x=1,3,7; mark which inputs are allowed over the reals.
5′ Explain $\sqrt{x^{2}}=|x|$ and the condition $x-3\geq0$ . A few examples suggest the rule; justify it using the definition of the nonnegative root.
2′ Individually simplify $\sqrt{(-5)^{2}}$ and distinguish it from solving $x^{2}=25$ .

Worked example1

Simplify exactly. State any restriction that is needed.

\sqrt{(-7)^{2}}

Reference answer

$7$

7

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
An intermediate form is $\sqrt{49}=7$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

Simplify exactly. State any restriction that is needed.

\sqrt{50}

Complete the square-factor split: $50=25\times\square$ .
Then $\sqrt{50}=\sqrt{25}\sqrt{\square}=\square\sqrt{2}$ .
Check that the coefficient is nonnegative and the square of the result is 50.

Reference answer

$5\sqrt{2}$

5\sqrt{2}

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\sqrt{25\times2}=5\sqrt{2}$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

Simplify exactly. State any restriction that is needed.

\sqrt{49}

Reference answer

$7$

7

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
The simplified form is $7$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

Is this expression defined over the real numbers? Explain.

\sqrt{-4}

Reference answer

No real value; the constant radicand −4 is negative.

\varnothing

Every radicand must be nonnegative.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $\varnothing$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Find all real values of x for which the expression is defined.

\sqrt{x-3}

Reference answer

$x\geq3$

x\geq3

Every radicand must be nonnegative.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $x\geq3$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

Find all real values of x for which the expression is defined.

\sqrt{5-2x}

Reference answer

$x\leq\frac{5}{2}$

x\leq\frac{5}{2}

Every radicand must be nonnegative.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $x\leq\frac{5}{2}$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

Simplify exactly. State any restriction that is needed.

\sqrt{x^{2}}

Reference answer

$|x|$

|x|

A square root cannot be negative, even when x is negative.
The simplified form is $|x|$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

Find all real values of x for which the expression is defined.

\frac{1}{\sqrt{x+2}}

Reference answer

$x>-2$

x>-2

Every radicand must be nonnegative; a square root in a denominator must be strictly positive.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $x>-2$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Simplify exactly. State any restriction that is needed.

\sqrt{0}

Reference answer

$0$

0

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
The simplified form is $0$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

A learner writes $\sqrt{25}=\pm5$ . Give the correct value and explain the error.

\sqrt{25}

Reference answer

$5$

5

The equation x²=25 has two solutions; the square-root symbol itself denotes only 5.
The simplified form is $5$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

Simplify exactly. State any restriction that is needed.

\sqrt{\frac{1}{4}}

Reference answer

$\frac{1}{2}$

\frac{1}{2}

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
The simplified form is $\frac{1}{2}$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

Find all real values of x for which the expression is defined.

\sqrt{x-1}+\sqrt{4-x}

Reference answer

$1\leq x\leq4$

1\leq x\leq4

Every radicand must be nonnegative.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $1\leq x\leq4$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

Simplify exactly. State any restriction that is needed.

\sqrt{(0-2)^{2}}

Reference answer

$2$

2

The square-root symbol gives the nonnegative root; a real radicand must be nonnegative.
An intermediate form is $|0-2|=2$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Simplify exactly. State any restriction that is needed.

(\sqrt{11})^{2}

Reference answer

$11$

11

Squaring the nonnegative square root restores its nonnegative radicand.
The simplified form is $11$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

Find all real values of x for which the expression is defined.

\frac{\sqrt{2x+1}}{\sqrt{3-x}}

Reference answer

$-\frac{1}{2}\leq x<3$

-\frac{1}{2}\leq x<3

Every radicand must be nonnegative; a square root in a denominator must be strictly positive.
Solve the resulting linear inequalities, then take their common part.
The complete permitted set is $-\frac{1}{2}\leq x<3$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

Use this value to disprove the claim $\sqrt{x^{2}}=x$ for every real x.

\sqrt{(-2)^{2}}

Reference answer

$2$

2

At x=−2, the root is 2 while x is −2. The correct identity uses absolute value.
The simplified form is $2$ .
Check the original expression and the sign of the square-root value.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.