PAPER · REASONING · PRACTICE

Square-root operations: factor, multiply and combine

Simplify square factors, apply real-domain product and quotient laws, then combine like radicals.

40 minutes · Year 8, lower volume · text pp. 5–12 · textbook lessons 3–6

Textbook reference: Year 8, lower volume · Chapter 16: Square-root expressions · lessons 3–6

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 whether adding square roots obeys the same rule as multiplying them. Give one test case. 8 min: On paper compare $\sqrt8\sqrt2=4$ with $\sqrt8+\sqrt2=3\sqrt2$ . Draw rectangles with sides √8 and √2 to interpret the product. Test $\sqrt9+\sqrt{16}$ against $\sqrt{9+16}$ . 5 min: Explain why 7≠5 refutes the proposed addition rule. State a,b≥0 for $\sqrt a\sqrt b=\sqrt{ab}$ ; combine only like radical terms after simplification. 2 min: Individually simplify $\sqrt{18}+\sqrt8$ and explain each transformation.

Organise the key idea

Only equal simplified radical parts can be combined by adding their coefficients.

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{12}+\sqrt{27}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $5\sqrt{3}$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $2\sqrt{3}+3\sqrt{3}$ .
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{18}\times\sqrt{8}

Both radicands are nonnegative, so $\sqrt{18}\sqrt{8}=\sqrt{18\times8}=\sqrt{\square}$ .
Find the nonnegative number satisfying $\square^{2}=144$ .
Check the product by simplifying both original roots separately.

A product of square roots with nonnegative radicands can be combined into one root.
18×8=144. Put that product under one square-root sign.
Choose the nonnegative square root of 144, not both signs of a squared equation.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\sqrt{144}=12$ .
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{16}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $4$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
The simplified form is $4$ .
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

Simplify exactly. State any restriction that is needed.

\sqrt{3}\times\sqrt{3}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $3$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
The simplified form is $3$ .
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 practice5

Simplify exactly. State any restriction that is needed.

\sqrt{75}-\sqrt{12}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $3\sqrt{3}$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $5\sqrt{3}-2\sqrt{3}$ .
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 practice6

Simplify exactly. State any restriction that is needed.

\frac{\sqrt{48}}{\sqrt{3}}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $4$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\sqrt{16}=4$ .
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 practice7

Simplify exactly. State any restriction that is needed.

\sqrt{2}(\sqrt{8}+\sqrt{18})

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $10$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $4+6=10$ .
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

Simplify exactly. State any restriction that is needed.

\frac{3}{\sqrt{3}}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $\sqrt{3}$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\frac{3\sqrt{3}}{3}$ .
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 question9

Simplify exactly. State any restriction that is needed.

\sqrt{20}+\sqrt{5}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $3\sqrt{5}$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $2\sqrt{5}+\sqrt{5}$ .
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.

Exit question10

Simplify exactly. State any restriction that is needed.

\frac{\sqrt{54}}{\sqrt{6}}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $3$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\sqrt{9}=3$ .
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.

2\sqrt{18}-\sqrt{8}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $4\sqrt{2}$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $6\sqrt{2}-2\sqrt{2}$ .
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.

Further practice12

Simplify exactly. State any restriction that is needed.

\sqrt{\frac{3}{4}}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $\frac{\sqrt{3}}{2}$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
The simplified form is $\frac{\sqrt{3}}{2}$ .
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 practice13

Simplify exactly. State any restriction that is needed.

(-2\sqrt{3})(3\sqrt{12})

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $-36$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $-6\sqrt{36}=-36$ .
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{2}+\sqrt{3}

The squarefree radical parts differ, so the two terms cannot be combined into one like-radical term.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $\sqrt{2}+\sqrt{3}$ and explain the simplification.

The squarefree radical parts differ, so the two terms cannot be combined into one like-radical term.
The simplified form is $\sqrt{2}+\sqrt{3}$ .
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

Simplify exactly. State any restriction that is needed.

(\sqrt{3}+\sqrt{2})^{2}

Extract square factors first. The square-root symbol denotes the nonnegative root.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $5+2\sqrt{6}$ and explain the simplification.

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $3+2\sqrt{6}+2$ .
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.

Extension16

Compare this value with $\sqrt{9}+\sqrt{16}$ to test whether square roots distribute over addition.

\sqrt{9+16}

The left value is 5; the proposed right value is 3+4=7, so the proposed rule fails.
Use products or quotients of square roots only under their real-domain conditions.
Compare your result with $5\neq7$ and explain the simplification.

The left value is 5; the proposed right value is 3+4=7, so the proposed rule fails.
The simplified form is $5\neq7$ .
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

Root meaning, signed fractions and the distributive law.

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. 5–12 · textbook lessons 3–6

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

Square-root operations: factor, multiply and combine

Year 8, lower volume · text pp. 5–12 · textbook lessons 3–6 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Simplify exactly. State any restriction that is needed.

\sqrt{12}+\sqrt{27}

Guided attempt2

Simplify exactly. State any restriction that is needed.

\sqrt{18}\times\sqrt{8}

Both radicands are nonnegative, so $\sqrt{18}\sqrt{8}=\sqrt{18\times8}=\sqrt{\square}$ .
Find the nonnegative number satisfying $\square^{2}=144$ .
Check the product by simplifying both original roots separately.

Prerequisite3

Simplify exactly. State any restriction that is needed.

\sqrt{16}

Prerequisite4

Simplify exactly. State any restriction that is needed.

\sqrt{3}\times\sqrt{3}

Square-root operations: factor, multiply and combine

Year 8, lower volume · text pp. 5–12 · textbook lessons 3–6 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Simplify exactly. State any restriction that is needed.

\sqrt{75}-\sqrt{12}

Independent practice6

Simplify exactly. State any restriction that is needed.

\frac{\sqrt{48}}{\sqrt{3}}

Independent practice7

Simplify exactly. State any restriction that is needed.

\sqrt{2}(\sqrt{8}+\sqrt{18})

Independent practice8

Simplify exactly. State any restriction that is needed.

\frac{3}{\sqrt{3}}

Exit question9

Simplify exactly. State any restriction that is needed.

\sqrt{20}+\sqrt{5}

Exit question10

Simplify exactly. State any restriction that is needed.

\frac{\sqrt{54}}{\sqrt{6}}

Square-root operations: factor, multiply and combine — Further practice

Year 8, lower volume · text pp. 5–12 · textbook lessons 3–6

Further practice11

Simplify exactly. State any restriction that is needed.

2\sqrt{18}-\sqrt{8}

Further practice12

Simplify exactly. State any restriction that is needed.

\sqrt{\frac{3}{4}}

Further practice13

Simplify exactly. State any restriction that is needed.

(-2\sqrt{3})(3\sqrt{12})

Further practice14

Simplify exactly. State any restriction that is needed.

\sqrt{2}+\sqrt{3}

Extension15

Simplify exactly. State any restriction that is needed.

(\sqrt{3}+\sqrt{2})^{2}

Extension16

Compare this value with $\sqrt{9}+\sqrt{16}$ to test whether square roots distribute over addition.

\sqrt{9+16}

Square-root operations: factor, multiply and combine — Teacher answers

Year 8, lower volume · text pp. 5–12 · textbook lessons 3–6

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: Root meaning, signed fractions and the distributive law.

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

3′ Predict whether adding square roots obeys the same rule as multiplying them. Give one test case.
8′ On paper compare $\sqrt8\sqrt2=4$ with $\sqrt8+\sqrt2=3\sqrt2$ . Draw rectangles with sides √8 and √2 to interpret the product. Test $\sqrt9+\sqrt{16}$ against $\sqrt{9+16}$ .
5′ Explain why 7≠5 refutes the proposed addition rule. State a,b≥0 for $\sqrt a\sqrt b=\sqrt{ab}$ ; combine only like radical terms after simplification.
2′ Individually simplify $\sqrt{18}+\sqrt8$ and explain each transformation.

Worked example1

Simplify exactly. State any restriction that is needed.

\sqrt{12}+\sqrt{27}

Reference answer

$5\sqrt{3}$

5\sqrt{3}

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $2\sqrt{3}+3\sqrt{3}$ .
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{18}\times\sqrt{8}

Both radicands are nonnegative, so $\sqrt{18}\sqrt{8}=\sqrt{18\times8}=\sqrt{\square}$ .
Find the nonnegative number satisfying $\square^{2}=144$ .
Check the product by simplifying both original roots separately.

Reference answer

$12$

12

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\sqrt{144}=12$ .
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{16}

Reference answer

$4$

4

Extract square factors first. The square-root symbol denotes the nonnegative root.
The simplified form is $4$ .
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

Simplify exactly. State any restriction that is needed.

\sqrt{3}\times\sqrt{3}

Reference answer

$3$

3

Extract square factors first. The square-root symbol denotes the nonnegative root.
The simplified form is $3$ .
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 practice5

Simplify exactly. State any restriction that is needed.

\sqrt{75}-\sqrt{12}

Reference answer

$3\sqrt{3}$

3\sqrt{3}

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $5\sqrt{3}-2\sqrt{3}$ .
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 practice6

Simplify exactly. State any restriction that is needed.

\frac{\sqrt{48}}{\sqrt{3}}

Reference answer

$4$

4

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\sqrt{16}=4$ .
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 practice7

Simplify exactly. State any restriction that is needed.

\sqrt{2}(\sqrt{8}+\sqrt{18})

Reference answer

$10$

10

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $4+6=10$ .
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

Simplify exactly. State any restriction that is needed.

\frac{3}{\sqrt{3}}

Reference answer

$\sqrt{3}$

\sqrt{3}

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\frac{3\sqrt{3}}{3}$ .
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 question9

Simplify exactly. State any restriction that is needed.

\sqrt{20}+\sqrt{5}

Reference answer

$3\sqrt{5}$

3\sqrt{5}

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $2\sqrt{5}+\sqrt{5}$ .
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.

Exit question10

Simplify exactly. State any restriction that is needed.

\frac{\sqrt{54}}{\sqrt{6}}

Reference answer

$3$

3

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $\sqrt{9}=3$ .
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.

2\sqrt{18}-\sqrt{8}

Reference answer

$4\sqrt{2}$

4\sqrt{2}

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $6\sqrt{2}-2\sqrt{2}$ .
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.

Further practice12

Simplify exactly. State any restriction that is needed.

\sqrt{\frac{3}{4}}

Reference answer

$\frac{\sqrt{3}}{2}$

\frac{\sqrt{3}}{2}

Extract square factors first. The square-root symbol denotes the nonnegative root.
The simplified form is $\frac{\sqrt{3}}{2}$ .
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 practice13

Simplify exactly. State any restriction that is needed.

(-2\sqrt{3})(3\sqrt{12})

Reference answer

$-36$

-36

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $-6\sqrt{36}=-36$ .
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{2}+\sqrt{3}

Reference answer

$\sqrt{2}+\sqrt{3}$

\sqrt{2}+\sqrt{3}

The squarefree radical parts differ, so the two terms cannot be combined into one like-radical term.
The simplified form is $\sqrt{2}+\sqrt{3}$ .
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

Simplify exactly. State any restriction that is needed.

(\sqrt{3}+\sqrt{2})^{2}

Reference answer

$5+2\sqrt{6}$

5+2\sqrt{6}

Extract square factors first. The square-root symbol denotes the nonnegative root.
An intermediate form is $3+2\sqrt{6}+2$ .
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.

Extension16

Compare this value with $\sqrt{9}+\sqrt{16}$ to test whether square roots distribute over addition.

\sqrt{9+16}

Reference answer

$5\neq7$

5\neq7

The left value is 5; the proposed right value is 3+4=7, so the proposed rule fails.
The simplified form is $5\neq7$ .
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.