PAPER · REASONING · PRACTICE

Square roots and cube roots: meanings and signs

Distinguish two square roots from one principal square root, and allow negative cube roots.

40 minutes · Year 7, lower volume · text pp. 29–34 · textbook lessons 15–17

Textbook reference: Year 7 lower · Chapter 6 Real numbers · text lessons 15–17

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

18 minutes: in the roots lab compare square areas 9, 2 and 0, predicting the allowed physical side lengths before checking (3 min prediction, 6 min trials). For area 9, contrast side length 3 with both solutions of x²=9; for area 0 explain why there is one value (4 min). On paper change the equation to x³=−8: find a negative cube root, and explain why this is not a negative square side (3 min). Sort the three requests “square roots”, “principal square root”, “cube root” and state the sign rules (2 min). The lab itself models only nonnegative square roots.

Organise the key idea

Distinguish two square roots from one principal square root, and allow negative cube roots.

Open the original chapter Open the interactive inquiry

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

Find all square roots of $81$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
A positive number has two opposite square roots.
Raise each proposed answer to the required power and check the original radicand.

For a positive radicand, both opposite values have the required square.
Check $(9)^{2}=81$ . The negative of this nonzero root has the same square.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Evaluate $\sqrt[3]{-64}$ .

Find r with $r^{3}=-64$ ; its sign is ____.
Complete $(-\square)^{3}=-64$ .
Therefore $\sqrt[3]{-64}=\square$ ; cube it to check.

This is a cube root, so a negative radicand is allowed.
A negative number cubed is negative; find a positive integer whose cube is 64.
Try the integers between 3 and 5, attach the negative sign and verify all three factors.

A cube root cubes to the given number.
Check $(-4)^{3}=-64$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

Evaluate $\sqrt{49}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
The principal square root cannot be negative.
Raise each proposed answer to the required power and check the original radicand.

The radical symbol denotes the nonnegative square root.
Check $(7)^{2}=49$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Evaluate $\sqrt{0}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
The principal square root cannot be negative.
Raise each proposed answer to the required power and check the original radicand.

The radical symbol denotes the nonnegative square root.
Check $(0)^{2}=0$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Evaluate $\sqrt{\frac{25}{36}}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
The principal square root cannot be negative.
Raise each proposed answer to the required power and check the original radicand.

The radical symbol denotes the nonnegative square root.
Check $(\frac{5}{6})^{2}=\frac{25}{36}$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

Find all square roots of $\frac{9}{100}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
A positive number has two opposite square roots.
Raise each proposed answer to the required power and check the original radicand.

For a positive radicand, both opposite values have the required square.
Check $(\frac{3}{10})^{2}=\frac{9}{100}$ . The negative of this nonzero root has the same square.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Evaluate $\sqrt[3]{\frac{-27}{125}}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
A negative radicand has a negative cube root.
Raise each proposed answer to the required power and check the original radicand.

A cube root cubes to the given number.
Check $(\frac{-3}{5})^{3}=\frac{-27}{125}$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Evaluate $\sqrt{(-8)^{2}}$ . Explain why the result is not −8.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

First (−8)²=64.
The nonnegative square root of 64 is 8; in general √(a²)=|a|.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

Does −9 have a real square root?

Yes, −3.
Yes, both 3 and −3.
No; a real square cannot be negative.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Both 3² and (−3)² are positive 9.
No real number has square −9.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Evaluate $\sqrt[3]{-8}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
A negative radicand has a negative cube root.
Raise each proposed answer to the required power and check the original radicand.

A cube root cubes to the given number.
Check $(-2)^{3}=-8$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Evaluate $\sqrt{100}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
The principal square root cannot be negative.
Raise each proposed answer to the required power and check the original radicand.

The radical symbol denotes the nonnegative square root.
Check $(10)^{2}=100$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

Find all square roots of $\frac{4}{49}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
A positive number has two opposite square roots.
Raise each proposed answer to the required power and check the original radicand.

For a positive radicand, both opposite values have the required square.
Check $(\frac{2}{7})^{2}=\frac{4}{49}$ . The negative of this nonzero root has the same square.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Evaluate $\sqrt[3]{216}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
A negative radicand has a negative cube root.
Raise each proposed answer to the required power and check the original radicand.

A cube root cubes to the given number.
Check $(6)^{3}=216$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Evaluate $\sqrt[3]{\frac{1}{64}}$ .

Identify whether the task asks for square roots, a principal square root, or a cube root.
A negative radicand has a negative cube root.
Raise each proposed answer to the required power and check the original radicand.

A cube root cubes to the given number.
Check $(\frac{1}{4})^{3}=\frac{1}{64}$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

For which real x is $\sqrt{x-3}$ defined? Include the endpoint.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

A real square-root radicand must be nonnegative: x−3≥0.
Thus x≥3; equality gives √0=0.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

Solve $x^{2}=12$ exactly. Explain how this differs from writing $\sqrt{12}$ .

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Both the positive and negative square roots square to 12.
The equation has two solutions; √12 names just one of them.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Preparation and assessment notes

Prerequisites

Calculate squares and cubes, including signed numbers and simple fractions.

Look for in students’ work

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

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 40-minute core consolidation after the mapped textbook lessons; not a replacement for every first-teaching period. Only the supplied text has been mapped.

大湾区初中数学三维课堂（七年级下册）正文.pdf

Year 7, lower volume · text pp. 29–34 · textbook lessons 15–17

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

Square roots and cube roots: meanings and signs

Year 7, lower volume · text pp. 29–34 · textbook lessons 15–17 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Find all square roots of $81$ .

Guided attempt2

Evaluate $\sqrt[3]{-64}$ .

Find r with $r^{3}=-64$ ; its sign is ____.
Complete $(-\square)^{3}=-64$ .
Therefore $\sqrt[3]{-64}=\square$ ; cube it to check.

Prerequisite3

Evaluate $\sqrt{49}$ .

Prerequisite4

Evaluate $\sqrt{0}$ .

Square roots and cube roots: meanings and signs

Year 7, lower volume · text pp. 29–34 · textbook lessons 15–17 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Evaluate $\sqrt{\frac{25}{36}}$ .

Independent practice6

Find all square roots of $\frac{9}{100}$ .

Independent practice7

Evaluate $\sqrt[3]{\frac{-27}{125}}$ .

Independent practice8

Evaluate $\sqrt{(-8)^{2}}$ . Explain why the result is not −8.

Exit question9

Does −9 have a real square root?

Yes, −3.
Yes, both 3 and −3.
No; a real square cannot be negative.

Exit question10

Evaluate $\sqrt[3]{-8}$ .

Square roots and cube roots: meanings and signs — Further practice

Year 7, lower volume · text pp. 29–34 · textbook lessons 15–17

Further practice11

Evaluate $\sqrt{100}$ .

Further practice12

Find all square roots of $\frac{4}{49}$ .

Further practice13

Evaluate $\sqrt[3]{216}$ .

Further practice14

Evaluate $\sqrt[3]{\frac{1}{64}}$ .

Extension15

For which real x is $\sqrt{x-3}$ defined? Include the endpoint.

Extension16

Solve $x^{2}=12$ exactly. Explain how this differs from writing $\sqrt{12}$ .

Square roots and cube roots: meanings and signs — Teacher answers

Year 7, lower volume · text pp. 29–34 · textbook lessons 15–17

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

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: Calculate squares and cubes, including signed numbers and simple fractions.

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

Inquiry and discussion: 18 minutes: in the roots lab compare square areas 9, 2 and 0, predicting the allowed physical side lengths before checking (3 min prediction, 6 min trials). For area 9, contrast side length 3 with both solutions of x²=9; for area 0 explain why there is one value (4 min). On paper change the equation to x³=−8: find a negative cube root, and explain why this is not a negative square side (3 min). Sort the three requests “square roots”, “principal square root”, “cube root” and state the sign rules (2 min). The lab itself models only nonnegative square roots.

Worked example1

Find all square roots of $81$ .

Reference answer

$\pm 9$ .

\pm 9

For a positive radicand, both opposite values have the required square.
Check $(9)^{2}=81$ . The negative of this nonzero root has the same square.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Evaluate $\sqrt[3]{-64}$ .

Find r with $r^{3}=-64$ ; its sign is ____.
Complete $(-\square)^{3}=-64$ .
Therefore $\sqrt[3]{-64}=\square$ ; cube it to check.

Reference answer

$-4$ .

-4

A cube root cubes to the given number.
Check $(-4)^{3}=-64$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

Evaluate $\sqrt{49}$ .

Reference answer

$7$ .

7

The radical symbol denotes the nonnegative square root.
Check $(7)^{2}=49$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Evaluate $\sqrt{0}$ .

Reference answer

$0$ .

0

The radical symbol denotes the nonnegative square root.
Check $(0)^{2}=0$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Evaluate $\sqrt{\frac{25}{36}}$ .

Reference answer

$\frac{5}{6}$ .

\frac{5}{6}

The radical symbol denotes the nonnegative square root.
Check $(\frac{5}{6})^{2}=\frac{25}{36}$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

Find all square roots of $\frac{9}{100}$ .

Reference answer

$\pm \frac{3}{10}$ .

\pm \frac{3}{10}

For a positive radicand, both opposite values have the required square.
Check $(\frac{3}{10})^{2}=\frac{9}{100}$ . The negative of this nonzero root has the same square.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Evaluate $\sqrt[3]{\frac{-27}{125}}$ .

Reference answer

$\frac{-3}{5}$ .

\frac{-3}{5}

A cube root cubes to the given number.
Check $(\frac{-3}{5})^{3}=\frac{-27}{125}$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Evaluate $\sqrt{(-8)^{2}}$ . Explain why the result is not −8.

Reference answer

$8$ ; a principal square root is nonnegative.

8

First (−8)²=64.
The nonnegative square root of 64 is 8; in general √(a²)=|a|.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

Does −9 have a real square root?

Yes, −3.
Yes, both 3 and −3.
No; a real square cannot be negative.

Reference answer

No; a real square cannot be negative.

Both 3² and (−3)² are positive 9.
No real number has square −9.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Evaluate $\sqrt[3]{-8}$ .

Reference answer

$-2$ .

-2

A cube root cubes to the given number.
Check $(-2)^{3}=-8$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Evaluate $\sqrt{100}$ .

Reference answer

$10$ .

10

The radical symbol denotes the nonnegative square root.
Check $(10)^{2}=100$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

Find all square roots of $\frac{4}{49}$ .

Reference answer

$\pm \frac{2}{7}$ .

\pm \frac{2}{7}

For a positive radicand, both opposite values have the required square.
Check $(\frac{2}{7})^{2}=\frac{4}{49}$ . The negative of this nonzero root has the same square.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Evaluate $\sqrt[3]{216}$ .

Reference answer

$6$ .

6

A cube root cubes to the given number.
Check $(6)^{3}=216$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Evaluate $\sqrt[3]{\frac{1}{64}}$ .

Reference answer

$\frac{1}{4}$ .

\frac{1}{4}

A cube root cubes to the given number.
Check $(\frac{1}{4})^{3}=\frac{1}{64}$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

For which real x is $\sqrt{x-3}$ defined? Include the endpoint.

Reference answer

$x\geq3$ .

x\geq3

A real square-root radicand must be nonnegative: x−3≥0.
Thus x≥3; equality gives √0=0.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

Solve $x^{2}=12$ exactly. Explain how this differs from writing $\sqrt{12}$ .

Reference answer

$x=\pm\sqrt{12}$ ; the radical alone is the nonnegative root.

x=\pm\sqrt{12}

Both the positive and negative square roots square to 12.
The equation has two solutions; √12 names just one of them.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.