Squares and Square Roots — Exercise 3.4

Find the square root of the following by division method:

2401

Answer

Square root of 2401 by division method:

Hence, $\mathbf{\sqrt{2401} = 49}$

Find the square root of the following by division method:

4489

Answer

Square root of 4489 by division method:

Hence, $\mathbf{\sqrt{4489} = 67}$

Find the square root of the following by division method:

106929

Answer

Square root of 106929 by division method:

Hence, $\mathbf{\sqrt{106929} = 327}$

Find the square root of the following by division method:

167281

Answer

Square root of 167281 by division method:

Hence, $\mathbf{\sqrt{167281} = 409}$

Find the square root of the following by division method:

53824

Answer

Square root of 53824 by division method:

Hence, $\mathbf{\sqrt{53824} = 232}$

Find the square root of the following by division method:

213444

Answer

Square root of 213444 by division method:

Hence, $\mathbf{\sqrt{213444} = 462}$

Find the square root of the following decimal numbers by division method:

51.84

Answer

Square root of 51.84 by division method:

Hence, $\mathbf{\sqrt{51.84} = 7.2}$

Find the square root of the following decimal numbers by division method:

42.25

Answer

Square root of 42.25 by division method:

Hence, $\mathbf{\sqrt{42.25} = 6.5}$

Find the square root of the following decimal numbers by division method:

18.4041

Answer

Square root of 18.4041 by division method:

Hence, $\mathbf{\sqrt{18.4041} = 4.29}$

Find the square root of the following decimal numbers by division method:

5.774409

Answer

Square root of 5.774409 by division method:

Hence, $\mathbf{\sqrt{5.774409} = 2.403}$

Find the square root of the following numbers correct to two decimal places:

645.8

Answer

Square root of 645.8 correct to two decimal places by division method:

$\sqrt{645.8} = 25.412 \approx 25.41$

Hence, $\mathbf{\sqrt{645.8} = 25.41}$

Find the square root of the following numbers correct to two decimal places:

107.45

Answer

Square root of 107.45 correct to two decimal places by division method:

$\sqrt{107.45} = 10.365 \approx 10.36$

Hence, $\mathbf{\sqrt{107.45} = 10.36}$

Find the square root of the following numbers correct to two decimal places:

5.462

Answer

Square root of 5.462 correct to two decimal places by division method:

$\sqrt{5.462} = 2.337 \approx 2.34$

Hence, $\mathbf{\sqrt{5.462} = 2.34}$

Find the square root of the following numbers correct to two decimal places:

2

Answer

Square root of 2 correct to two decimal places by division method:

$\sqrt{2} = 1.414 \approx 1.41$

Hence, $\mathbf{\sqrt{2} = 1.41}$

Find the square root of the following numbers correct to two decimal places:

3

Answer

Square root of 3 correct to two decimal places by division method:

$\sqrt{3} = 1.732 \approx 1.73$

Hence, $\mathbf{\sqrt{3} = 1.73}$

Find the square root of the following fractions correct to two decimal places:

$11\dfrac{3}{8}$

Answer

Converting mixed fraction into decimal:

$\Rightarrow \sqrt{11\dfrac{3}{8}}\\[1em] \Rightarrow \sqrt{\dfrac{91}{8}}\\[1em] \Rightarrow \sqrt{11.375}$

Square root of 11.375 correct to two decimal places by division method:

$\sqrt{11.375} = 3.372 \approx 3.37$

Hence, $\mathbf{\sqrt{11\dfrac{3}{8}} = 3.37}$

Find the square root of the following fractions correct to two decimal places:

$5\dfrac{5}{11}$

Answer

Converting mixed fraction into decimal:

$\Rightarrow \sqrt{5\dfrac{5}{11}}\\[1em] \Rightarrow \sqrt{\dfrac{60}{11}}\\[1em] \Rightarrow \sqrt{5.4545}$

Square root of 5.4545 correct to two decimal places by division method:

$\sqrt{5.4545} = 2.335 \approx 2.34$

Hence, $\mathbf{\sqrt{5\dfrac{5}{11}} = 2.34}$

Find the square root of the following fractions correct to two decimal places:

$7\dfrac{1}{3}$

Answer

Converting mixed fraction into decimal:

$\Rightarrow \sqrt{7\dfrac{1}{3}}\\[1em] \Rightarrow \sqrt{\dfrac{22}{3}}\\[1em] \Rightarrow \sqrt{7.3333}$

Square root of 7.3333 correct to two decimal places by division method:

$\sqrt{7.3333} = 2.708 \approx 2.71$

Hence, $\mathbf{\sqrt{7\dfrac{1}{3}} = 2.71}$

Find the least number which must be subtracted from the following number to make it a perfect square. Also find the square root of the perfect square number so obtained:

2000

Answer

Using division method:

Clearly, if 64 is subtracted from 2000, the remainder will be a perfect square.

Since, 2000 - 64 = 1936

and, $\sqrt{1936} = 44$

Hence, 64 should be subtracted from 2000 and $\mathbf{\sqrt{1936} = 44}$.

Find the least number which must be subtracted from the following number to make it a perfect square. Also find the square root of the perfect square number so obtained:

984

Answer

Using division method:

Clearly, if 23 is subtracted from 984, the remainder will be a perfect square.

Since, 984 - 23 = 961

and, $\sqrt{961} = 31$

Hence, 23 should be subtracted from 984 and $\mathbf{\sqrt{961} = 31}$.

Find the least number which must be subtracted from the following number to make it a perfect square. Also find the square root of the perfect square number so obtained:

8934

Answer

Using division method:

Clearly, if 98 is subtracted from 8934, the remainder will be a perfect square.

Since, 8934 - 98 = 8836

and, $\sqrt{8836} = 94$

Hence, 98 should be subtracted from 8934 and $\mathbf{\sqrt{8836} = 94}$.

Find the least number which must be subtracted from the following number to make it a perfect square. Also find the square root of the perfect square number so obtained:

11021

Answer

Using division method:

Clearly, if 205 is subtracted from 11021, the remainder will be a perfect square.

Since, 11021 - 205 = 10816

and, $\sqrt{10816} = 104$

Hence, 205 should be subtracted from 11021 and $\mathbf{\sqrt{10816} = 104}$.

Find the least number which must be added to the following number to make it a perfect square. Also find the square root of the perfect square number so obtained:

1750

Answer

Using division method:

Clearly, 1750 is greater than 41².

∴ On adding the required number to 1750, we shall be getting 42² i.e, 1764.

So, the required number = 1764 - 1750 = 14.

and, $\sqrt{1764} = 42$

Hence, 14 should be added to 1750 and $\mathbf{\sqrt{1764} = 42}$.

Find the least number which must be added to the following number to make it a perfect square. Also find the square root of the perfect square number so obtained:

6412

Answer

Using division method:

Clearly, 6412 is greater than 80².

∴ On adding the required number to 6412, we shall be getting 81² i.e, 6561.

So, the required number = 6561 - 6412 = 149.

and, $\sqrt{6561} = 81$

Hence, 149 should be added to 6412 and $\mathbf{\sqrt{6561} = 81}$.

Find the least number which must be added to the following number to make it a perfect square. Also find the square root of the perfect square number so obtained:

6598

Answer

Using division method:

Clearly, 6598 is greater than 81².

∴ On adding the required number to 6598, we shall be getting 82² i.e, 6724.

So, the required number = 6724 - 6598 = 126.

and, $\sqrt{6724} = 82$

Hence, 126 should be added to 6598 and $\mathbf{\sqrt{6724} = 82}$.

Find the least number which must be added to the following number to make it a perfect square. Also find the square root of the perfect square number so obtained:

8000

Answer

Using division method:

Clearly, 8000 is greater than 89².

∴ On adding the required number to 8000, we shall be getting 90² i.e, 8100.

So, the required number = 8100 - 8000 = 100.

and, $\sqrt{8100} = 90$

Hence, 100 should be added to 8000 and $\mathbf{\sqrt{8100} = 90}$.

Find the smallest four digit number which is a perfect square.

Answer

The smallest four digit number = 1000

Using division method:

As we can see 1000 is not a perfect square.

Clearly, 1000 is greater than 31².

∴ On adding the required number to 1000, we shall be getting 32² i.e, 1024.

So, the required number = 1024 - 1000 = 24.

1000 + 24 = 1024 is a perfect square.

Hence, the smallest four digit number which is a perfect square is 1024.

Find the greatest number of six digits which is a perfect square.

Answer

The greatest six digit number = 999999

Using division method:

We get the remainder = 1998

It means (999)² is less than 999999 by 1998.

∴ If we subtract 1998 from 999999, we get (999)² which is a perfect square number.

So, the required perfect square number = 999999 - 1998 = 998001.

Hence, the greatest six digit number which is a perfect square is 998001.

In a right triangle ABC, ∠B = 90°.

If AB = 14 cm, BC = 48 cm, find AC

Answer

In right triangle ABC, ∠B = 90°.

By Pythagoras theorem,

$\Rightarrow AC^2 = AB^2 + BC^2\\[1em] \Rightarrow AC^2 = 14^2 + 48^2\\[1em] \Rightarrow AC^2 = 196 + 2304\\[1em] \Rightarrow AC^2 = 2500\\[1em] \Rightarrow AC = \sqrt{2500}\\[1em] \Rightarrow AC = 50$

Hence, AC = 50 cm.

In a right triangle ABC, ∠B = 90°.

If AC = 37 cm, BC = 35 cm, find AB.

Answer

In right triangle ABC, ∠B = 90°.

By Pythagoras theorem,

$\Rightarrow AC^2 = AB^2 + BC^2\\[1em] \Rightarrow AB^2 = AC^2 - BC^2\\[1em] \Rightarrow AB^2 = 37^2 - 35^2\\[1em] \Rightarrow AB^2 = 1369 - 1225\\[1em] \Rightarrow AB^2 = 144\\[1em] \Rightarrow AB = \sqrt{144}\\[1em] \Rightarrow AB = 12$

Hence, AB = 12 cm.

A gardener has 1400 plants. He wants to plant these in such a way that the number of rows and number of columns remains same. Find the minimum number of plants he needs more for this.

Answer

Using division method:

Clearly, 1400 is greater than 37².

∴ On adding the required number to 1400, we shall be getting 38² i.e, 1444.

So, the required number = 1444 - 1400 = 44.

Hence, the gardener needs 44 more plants.

There are 1000 children in a school. For a P.T. drill they have to stand in such a way that the number of rows is equal to number of columns. How many children would be left out in this arrangement?

Answer

Using division method:

We get the remainder = 39

It means (31)² is less than 1000 by 39.

∴ If 39 children are left out, the remaining 961 children can be arranged in 31 rows and 31 columns.

Hence, 39 children would be left out in this arrangement.

Amit walks 16 m south from his house and turns east to walk 63 m to reach his friend's house. While returning, he walks diagonally from his friend's house to reach back to his house. What distance did he walk while returning?

Answer

Let A be the house of Amit, B the point where he turns east and C be his friend's house.

AB = 16 m, BC = 63 m and ∠B = 90°.

By Pythagoras theorem,

$\Rightarrow AC^2 = AB^2 + BC^2\\[1em] \Rightarrow AC^2 = 16^2 + 63^2\\[1em] \Rightarrow AC^2 = 256 + 3969\\[1em] \Rightarrow AC^2 = 4225\\[1em] \Rightarrow AC = \sqrt{4225}\\[1em] \Rightarrow AC = 65$

Hence, Amit walked a distance of 65 m while returning.

A ladder 6 m long leaned against a wall. The ladder reaches the wall to a height of 4.8 m. Find the distance between the wall and the foot of the ladder.

Answer

Let AC be the ladder, AB the wall and BC the distance between the wall and the foot of the ladder.

AC = 6 m, AB = 4.8 m and ∠B = 90°.

By Pythagoras theorem,

$\Rightarrow AC^2 = AB^2 + BC^2\\[1em] \Rightarrow BC^2 = AC^2 - AB^2\\[1em] \Rightarrow BC^2 = 6^2 - (4.8)^2\\[1em] \Rightarrow BC^2 = 36 - 23.04\\[1em] \Rightarrow BC^2 = 12.96\\[1em] \Rightarrow BC = \sqrt{12.96}\\[1em] \Rightarrow BC = 3.6$

Hence, the distance between the wall and the foot of the ladder is 3.6 m.