ICSE Class 8 Maths Number Systems Chapter 09 Squares and Square Roots

Squares And Square Roots

Square of a Number

Square Root of a Number

Square Roots of Perfect Squares

Square Roots of Non-Perfect Squares

Square Of A Number

The square of a number is the number multiplied by itself. The number is called the base and as it is multiplied by itself only once, its power or index is 2.

\(a^2 = a \times a\), base = a, power = 2

Example 1: \(18^2 = 18 \times 18 = 324\)

Example 2: \(291^2 = 291 \times 291 = 84681\)

Example 3: \(\left(\frac{5}{7}\right)^2 = \frac{5}{7} \times \frac{5}{7} = \frac{25}{49}\)

Example 4: \(\left(\frac{8}{5}\right)^2 = \frac{8}{5} \times \frac{8}{5} = \frac{64}{25} = 2\frac{14}{25}\)

Example 5: \((0.79)^2 = 0.79 \times 0.79 = 0.6241\)

Example 6: \((1.3)^2 = 1.3 \times 1.3 = 1.69\)

Try this!

1. \(5^2 =\)

2. \(\left(\frac{4}{5}\right)^2 =\)

Observe from the above examples that

the square of an even number is an even number.

the square of an odd number is an odd number.

the square of a proper fraction is less than the proper fraction.

the square of an improper fraction is greater than the improper fraction.

Teacher's Note

When you calculate the area of a square room by multiplying its length by itself, you are finding the square of that measurement.

Numbers which can be expressed as the square of two exact rational numbers are known as perfect squares.

Example 7: Is 3528 a perfect square?

Express 3528 as a product of its prime factors.

\(3528 = 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7\)

Pairing off the factors we find one factor left unpaired.

\(3528 = 2 \times 42 \times 42\)

Thus, 3528 is not a perfect square.

Example 8: What is the smallest number by which 3528 should be divided to make it a perfect square?

\(3528 = 2 \times 42 \times 42\)

If 3528 is divided by 2, then 1764 = 42 \times 42, which is a perfect square.

Square Root Of A Number

When a number 'a' multiplied by itself gives a certain product \(a^2\), a is known as the square root of \(a^2\).

\(\sqrt{a^2} = \sqrt{a \times a} = a\), where the radical sign '\(\sqrt{\phantom{x}}\)' denotes square root.

Example 9: \(\sqrt{81} = \sqrt{9 \times 9} = 9\)

Example 10: \(\sqrt{400} = \sqrt{20 \times 20} = 20\)

Example 11: \(\sqrt{\frac{9}{16}} = \sqrt{\frac{3}{4} \times \frac{3}{4}} = \frac{3}{4}\)

Example 12: \(\sqrt{\frac{21}{100}} = \sqrt{\frac{121}{100}} = \sqrt{\frac{11}{10} \times \frac{11}{10}} = \frac{11}{10}\)

Example 13: \(\sqrt{0.49} = \sqrt{0.7 \times 0.7} = 0.7\)

Example 14: \(\sqrt{6.25} = \sqrt{2.5 \times 2.5} = 2.5\)

The square roots in all the examples above are exact rational numbers.

Try this!

1. \(\sqrt{225}\)

2. \(\sqrt{1024}\)

Square Roots Of Perfect Squares By Prime Factorisation Method

Step 1: Obtain the prime factorisation of the given number.

Step 2: Make pairs of identical factors.

Step 3: Find the product of one factor taken from each pair obtained above.

Example 15: Find the square root of 7056.

\(\sqrt{7056} = \sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7}\)

\(= 2 \times 2 \times 3 \times 7\)

\(= 84\)

Example 16: Find the square root of \(\sqrt[3]{\frac{109}{225}}\).

\(\sqrt[3]{\frac{109}{225}} = \sqrt{\frac{784}{225}} = \sqrt{\frac{2 \times 2 \times 2 \times 2 \times 7 \times 7}{3 \times 5 \times 5}}\)

\(= \frac{2 \times 2 \times 7}{3 \times 5} = \frac{28}{15} = 1\frac{13}{15}\)

Example 17: Find the square root of 6.76.

\(\sqrt{6.76} = \sqrt{\frac{676}{100}} = \sqrt{\frac{2 \times 2 \times 13 \times 13}{2 \times 2 \times 5 \times 5}}\)

\(= \frac{2 \times 13}{2 \times 5} = \frac{26}{10} = 2.6\)

Teacher's Note

Understanding perfect squares helps you quickly estimate measurements, like determining if a plot of land with a certain area can form a perfect square shape.

Exercise 9.1

1. Find the value of the following.

(i) \(14^2\) - (ii) \(27^2\)

(iii) \(118^2\) - (iv) \(\left(\frac{1}{5}\right)^2\)

(v) \(\left(\frac{3}{8}\right)^2\) - (vi) \(\left(1\frac{2}{5}\right)^2\)

(vii) \(\left(2\frac{4}{5}\right)^2\) - (viii) \((0.1)^2\)

(ix) \((5.8)^2\) - (x) \((3.05)^2\)

2. Which of the following perfect squares will have even square roots and which will have odd square roots?

(i) 441 - (ii) 2916

(iii) 3969 - (iv) 21609

(v) 389376

3. Which of the following numbers are perfect squares?

(i) 128 - (ii) 256

(iii) 2187 - (iv) 6561

(v) 6084

4. Find the perfect square obtained by multiplying each of the following numbers by the smallest possible number.

(i) 882 - (ii) 405

(iii) 567 - (iv) 1690

(v) 7776

5. Divide the following numbers by the smallest possible number to make each one a perfect square.

(i) 1152 - (ii) 2187

(iii) 3267 - (iv) 1536

(v) 10140

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