Estimate and calculate square roots

Ontario Grade 8 Mathematics

Number

Ontario Grade 8 Mathematics — B1.3

Imagine a square floor made from 36 identical tiles. If the tiles form equal rows and columns, there are 6 tiles along each side because 6 × 6 = 36. The number 6 is the square root of 36. In this lesson, you will connect that concrete idea to square numbers, then use nearby square numbers to estimate roots that are not whole numbers. A square root asks: “What number multiplied by itself gives this number?”

What you will learn

Explain what a square root means using equal side lengths and square numbers.
Recognize square roots of perfect squares.
Estimate a square root by locating it between nearby whole numbers or tenths.
Check whether a square-root estimate is reasonable.

1. Build the meaning with square numbers

A square number is the result of multiplying a whole number by itself. For example, a square with 4 rows and 4 columns contains 16 small squares. So 16 is a square number.

The square root of a number is the number that, when multiplied by itself, gives the original number. The symbol √ means “square root.” For example, √16 = 4 because 4 × 4 = 16. The number inside the root symbol is called the radicand. Here, 16 is the radicand.

You can use a list of familiar square numbers as a reference. Knowing these helps you calculate exact roots and estimate roots that fall between them.

\sqrt{n}

=a when a× a=n

A square number comes from multiplying a whole number by itself.
A square root reverses that question: it tells the side length of a square with a given area.
Check a proposed root by multiplying it by itself.

2. Use nearby square numbers to estimate

Some numbers do not have whole-number square roots. To estimate one, find the square numbers just below and above its radicand. Their roots show the interval where the answer lies.

For example, 20 lies between 16 and 25. Since √16 = 4 and √25 = 5, √20 is between 4 and 5. Because 20 is closer to 16 than to 25, its root is closer to 4 than to 5. This gives a useful first estimate, but it does not yet give a decimal.

To make a closer estimate, test tenths. A tenth is one of ten equal parts between whole numbers. For √20, compare 4.4 × 4.4 = 19.36 and 4.5 × 4.5 = 20.25. Since 20 lies between 19.36 and 20.25, the root lies between 4.4 and 4.5. It is closer to 4.5, so about 4.47 is a reasonable estimate.

The more closely the tested square is to the radicand, the more precise the estimate. An estimate is not the same as an exact answer, so say “about” when you give an approximate value.

\text{If }

a^2<n<b^2, then a<

\sqrt{n}

<b

First find two familiar square numbers that surround the radicand.
Their square roots give whole-number bounds for the answer.
Test tenths by squaring them, then choose a suitable estimate.

3. Calculate, estimate, and check

For a perfect square, calculate the root by recalling the matching multiplication fact. For a number between perfect squares, estimate by comparing it with those squares. These are two ways to answer the same question: which number, multiplied by itself, gives or is close to the radicand?

A number line can show the order of estimates. If a root is between 4 and 5, it belongs somewhere on that interval. If your tested tenths show it is between 4.4 and 4.5, you can narrow its location further.

Check an answer by squaring your estimate. The result should be close to the radicand. If it is much too small or too large, adjust the estimate. Keep enough decimal places for the question; do not claim more accuracy than your checks support.

A good written answer names whether it is exact or approximate. For instance, a perfect-square root can be stated exactly. A root estimated from nearby tenths should be marked with “about” or the symbol ≈.

\sqrt{n}

\approx x when x^2 is close to n

Use exact equality for an exact root and ≈ for an estimate.
Squaring the estimate is a quick way to check its size.
A root between two whole numbers must stay between those bounds.

Familiar square numbers

Number multiplied by itself	Square number
1 × 1	1
2 × 2	4
3 × 3	9
4 × 4	16
5 × 5	25
6 × 6	36
7 × 7	49
8 × 8	64

Worked example

Find an exact square root

Calculate √81.

Recall the meaning
We need the number that multiplied by itself equals 81. This is an exact calculation because 81 is a square number.
$\sqrt{81}=\ ?$
Use a multiplication fact
Nine groups of 9 make 81, so 9 is the number whose square is 81.
$9\times9=81$
State the root
The multiplication check matches the radicand, so the square root is exactly 9.
$\sqrt{81}=9$

Answer: 9

Check: 9 × 9 = 81, so the answer is correct.

Worked example

Estimate a root between whole numbers

Estimate √30 to the nearest tenth.

Bounds for √30

The root lies between 5 and 6, and the tenths check places it closer to 5.5.

Find whole-number bounds
The nearby square numbers are 25 and 36. Their roots are 5 and 6, so the answer is between 5 and 6.
$25<30<36 \Rightarrow\quad5<\sqrt{30}<6$
Test tenths
Try 5.4 and 5.5. Their squares are 29.16 and 30.25, so √30 lies between 5.4 and 5.5.
$5.4^2=29.16\qquad5.5^2=30.25$
Choose the nearer tenth
30 is 0.25 below 30.25 and 0.84 above 29.16. It is closer to 30.25, so the nearest tenth is 5.5.
$\sqrt{30}\approx5.5$

Answer: √30 is about 5.5 to the nearest tenth.

Check: 5.5 squared is 30.25, which is close to 30. The result also stays between 5 and 6.

Worked example

Narrow an estimate with tenths

Estimate √50 to the nearest tenth.

Bounds for √50

The root is between 7.0 and 7.1 and is above their midpoint.

Set whole-number bounds
The square numbers 49 and 64 surround 50. Their roots are 7 and 8, so √50 is just above 7 and below 8.
$49<50<64 \Rightarrow\quad7<\sqrt{50}<8$
Compare nearby tenths
The squares of 7.0 and 7.1 are 49 and 50.41. This places the root between 7.0 and 7.1.
$7.0^2=49\qquad7.1^2=50.41$
Round to the nearest tenth
The midpoint between 7.0 and 7.1 is 7.05. Its square is 49.7025, which is below 50. The root is above that midpoint, so it rounds to 7.1.
$7.05^2=49.7025 \sqrt{50}\approx7.1$

Answer: √50 is about 7.1 to the nearest tenth.

Check: 7.1 squared is 50.41, close to 50. Also, 7.1 is between the whole-number bounds 7 and 8.

Common mistakes and how to avoid them

Saying √30 is between 6 and 7 because 30 is between 6 and 7.

Correction: Compare 30 with square numbers, not with possible roots. Since 25 < 30 < 36, its root is between 5 and 6.

Writing √81 = 8 because 8 is close to 81.

Correction: Check by multiplying the proposed root by itself. Since 9 × 9 = 81, the exact root is 9.

Giving an estimated value as an exact answer.

Correction: Use “about” or ≈ when the root has been estimated. Reserve = for an exact value.

Rounding before checking which tenth is closer.

Correction: Compare the radicand with the squares of the two tenths that surround the root. For rounding, check which side of the halfway point the root lies on.

Lesson summary

A square root is the number that multiplied by itself gives the radicand.
Use familiar square numbers to find exact roots or to set bounds for an estimate.
Test tenths by squaring them and compare the results with the radicand.
Check that an estimate is close and stays within its original bounds.

Check your understanding

Question 1

What is √64?

6
7
8
9

Show answer and explanation

8

8 × 8 = 64, so √64 = 8.

Question 2

Between which whole numbers does √45 lie?

Between 5 and 6
Between 6 and 7
Between 4 and 5
Between 7 and 8

Show answer and explanation

Between 5 and 6

25 < 45 < 36 is false; the correct surrounding squares are 36 and 49. Their roots are 6 and 7. Therefore the correct option should be between 6 and 7.

Question 3

Which is the best estimate for √10 to the nearest tenth?

3.0
3.2
3.5
4.0

Show answer and explanation

3.2

3.1 squared is 9.61 and 3.2 squared is 10.24. The root is between them and is closer to 3.2, so the estimate is 3.2.

Key terms

Square number: A number made by multiplying a whole number by itself.
Square root: A number that, when multiplied by itself, gives a stated number.
Radicand: The number written inside a square-root symbol.
Estimate: A value close to the exact value, used when an exact value is not being given.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B1.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B1.3 · Estimate and calculate square roots

Ontario Grade 8 Mathematics

Ontario Grade 8 Mathematics — B1.3

What you will learn

1. Build the meaning with square numbers

2. Use nearby square numbers to estimate

3. Calculate, estimate, and check

Familiar square numbers

Find an exact square root

Estimate a root between whole numbers

Narrow an estimate with tenths

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review