RD Sharma solution class 8 chapter 3 Square and Square roots Exercise 3.6

Exercise 3.6

Page-3.48

Question 1:

Find the square root of:
(i) $\frac{441}{961}$
(ii) $\frac{324}{841}$
(iii) $4\frac{29}{29}$
(iv) $2\frac{14}{25}$
(v) $2\frac{137}{196}$
(vi) $23\frac{26}{121}$
(vii) $25\frac{544}{729}$
(viii) $75\frac{46}{49}$
(ix) $3\frac{942}{2209}$
(x) $3\frac{334}{3025}$
(xi) $21\frac{2797}{3364}$
(xii) $38\frac{11}{25}$
(xiii) $23\frac{394}{729}$
(xiv) $21\frac{51}{169}$
(xv) $10\frac{151}{225}$

Answer 1:

(i) We know:
$\sqrt{\frac{441}{961}}=\frac{\sqrt{441}}{\sqrt{961}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$$\begin{gathered} \sqrt{441}=\sqrt{\left(3\times 3\right)\times \left(7\times 7\right)}=3\times 7=21 \\ \sqrt{961}=\sqrt{31\times 31}=31 \\ \therefore \sqrt{\frac{441}{961}}=\frac{21}{31} \end{gathered}$$

(ii)We know:
$\sqrt{\frac{324}{841}}=\frac{\sqrt{324}}{\sqrt{841}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$$\begin{gathered} \sqrt{324}=\sqrt{2\times 2\times 3\times 3\times 3\times 3}=2\times 3\times 3=18 \\ \sqrt{841}=\sqrt{29\times 29}=29 \\ \therefore \sqrt{\frac{324}{841}}=\frac{18}{29} \end{gathered}$$
(iii) By looking at the book's answer key, the fraction should be $\sqrt{4\frac{29}{49}}, \mathrm{not} \sqrt{4\frac{29}{29}}$.
We know:
$$\begin{gathered} \sqrt{4\frac{29}{49}}=\sqrt{\frac{225}{49}}=\frac{\sqrt{225}}{\sqrt{49}} \\ \sqrt{225}=15 \\ \sqrt{49}=7 \\ \therefore \sqrt{4\frac{29}{49}}=\frac{15}{7} \end{gathered}$$

(iv) We know:
$\sqrt{2\frac{14}{25}}=\sqrt{\frac{64}{25}}=\frac{\sqrt{64}}{\sqrt{25}}=\frac{8}{5}$

(v) We know:
$\sqrt{2\frac{137}{196}}=\sqrt{\frac{529}{196}}=\frac{\sqrt{529}}{\sqrt{196}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$$\begin{gathered} \sqrt{529}=\sqrt{23\times 23}=23 \\ \sqrt{196}=\sqrt{2\times 2\times 7\times 7}=2\times 7=14 \\ \therefore \sqrt{2\frac{137}{196}}=\frac{23}{14} \end{gathered}$$

(vi) We know:
$\sqrt{23\frac{26}{121}}=\sqrt{\frac{2809}{121}}=\frac{\sqrt{2809}}{\sqrt{121}}$
Now, let us compute the square roots of the numerator and the denominator separately.

$$\begin{gathered} \sqrt{121}=11 \\ \therefore \sqrt{23\frac{26}{121}}=\frac{53}{11} \end{gathered}$$

(vii) We know:
$\sqrt{25\frac{544}{729}}=\sqrt{\frac{18769}{729}}=\frac{\sqrt{18769}}{\sqrt{729}}$
Now, let us compute the square roots of the numerator and the denominator separately.

$$\begin{gathered} \sqrt{729}=27 \\ \therefore \sqrt{25\frac{544}{729}}=\frac{137}{27} \end{gathered}$$

(viii) We know:
$\sqrt{75\frac{46}{49}}=\sqrt{\frac{3721}{49}}=\frac{\sqrt{3721}}{\sqrt{49}}$
Now, let us compute the square roots of the numerator and the denominator separately.

$$\begin{gathered} \sqrt{49}=7 \\ \therefore \sqrt{75\frac{46}{49}}=\frac{61}{7} \end{gathered}$$

(ix) We know:
$\sqrt{3\frac{942}{2209}}=\sqrt{\frac{7569}{2209}}=\frac{\sqrt{7569}}{\sqrt{2209}}$
Now, let us compute the square roots of the numerator and the denominator separately.

$\therefore \sqrt{3\frac{942}{2209}}=\frac{87}{47}$

(x) We know:
$\sqrt{3\frac{334}{3025}}=\sqrt{\frac{9409}{3025}}=\frac{\sqrt{9409}}{\sqrt{3025}}$
Now, let us compute the square roots of the numerator and the denominator separately.

$\therefore \sqrt{3\frac{334}{3025}}=\frac{97}{55}$

(xi) We know:
$\sqrt{21\frac{2797}{3364}}=\sqrt{\frac{73441}{3364}}=\frac{\sqrt{73441}}{\sqrt{3364}}$
Now, let us compute the square roots of the numerator and the denominator separately.

$\therefore \sqrt{21\frac{2797}{3364}}=\frac{271}{58}$

(xii) We know:
$\sqrt{38\frac{11}{25}}=\sqrt{\frac{961}{25}}=\frac{\sqrt{961}}{\sqrt{25}}$
Now, let us compute the square roots of the numerator and the denominator separately.
$$\begin{gathered} \sqrt{961}=31 \\ \sqrt{25}=5 \\ \therefore \sqrt{38\frac{11}{25}}=\frac{31}{5} \end{gathered}$$

(xiii) We know:
$\sqrt{23\frac{394}{729}}=\sqrt{\frac{17161}{729}}=\frac{\sqrt{17161}}{\sqrt{729}}$
Now, let us compute the square roots of the numerator and the denominator separately.

$$\begin{gathered} \sqrt{729}=27 \\ \therefore \sqrt{23\frac{394}{729}}=\frac{131}{27}=4\frac{23}{27} \end{gathered}$$

(xiv) We know:
$\sqrt{21\frac{51}{169}}=\sqrt{\frac{3600}{169}}=\frac{\sqrt{3600}}{169}$
Now, let us compute the square roots of the numerator and the denominator separately.
$$\begin{gathered} \sqrt{3600}=\sqrt{60\times 60}=60 \\ \sqrt{169}=\sqrt{13\times 13}=13 \\ \therefore \sqrt{21\frac{51}{169}}=\frac{60}{13}=4\frac{8}{13} \end{gathered}$$

(xv) We know:
$\sqrt{10\frac{151}{225}}=\sqrt{\frac{2401}{225}}=\frac{\sqrt{2401}}{\sqrt{225}}$
Now let us compute the square roots of the numerator and the denominator separately.
$$\begin{gathered} \sqrt{2401}=\sqrt{7\times 7\times 7\times 7}=7\times 7=49 \\ \sqrt{225}=\sqrt{3\times 3\times 5\times 5}=3\times 5=15 \\ \therefore \sqrt{10\frac{151}{225}}=\frac{49}{15}=3\frac{4}{15} \end{gathered}$$

Question 2:

Find the value of:
(i) $\frac{\sqrt{80}}{\sqrt{405}}$
(ii) $\frac{\sqrt{441}}{\sqrt{625}}$
(iii) $\frac{\sqrt{1587}}{\sqrt{1728}}$
(iv) $\sqrt{72}\times \sqrt{338}$
(v) $\sqrt{45}\times \sqrt{20}$

Answer 2:

(i) We have:


(ii) Computing the square roots:


$\therefore$

(iii) We have:
     (by dividing both numbers by 3)
Computing the square roots of the numerator and the denominator:


$\therefore$

(iv) We have:

= 
                                                       
(v) We have:

                         
                          = 30

Question 3:

The area of a square field is $80\frac{244}{729}$ square metres. Find the length of each side of the field.

Answer 3:

The length of one side is the square root of the area of the field. Hence, we need to calculate the value of
We have

Now, to calculate the square root of the numerator and the denominator:

We know that:


$\mathrm{Therefore}, \mathrm{length} \mathrm{of} \mathrm{one} \mathrm{side} \mathrm{of} \mathrm{the} \mathrm{field} = \frac{242}{27} = 8\frac{26}{27} m$

Page-3.49

Question 4:

The area of a square field is $30\frac{1}{4}{\mathrm{m}}^2.$ Calculate the length of the side of the square.

Answer 4:

The length of one side is equal to the square root of the area of the field. Hence, we just need to calculate the value of .
We have:


Now, calculating the square root of the numerator and the denominator:



$\mathrm{Therefore}, \mathrm{the} \mathrm{length} \mathrm{of} \mathrm{the} \mathrm{side} \mathrm{of} \mathrm{the} \mathrm{square} = \sqrt{30\frac{1}{4}}= \frac{11}{2} = 5\frac{1}{2} \mathrm{m}$

Question 5:

Find the length of a side of a square playground whose area is equal to the area of a rectangular field of diamensions 72 m and 338 m.

Answer 5:

The area of the playground = 72 $\times$ 338 = 24336 m²
The length of one side of a square is equal to the square root of its area. Hence, we just need to find the square root of 24336.

Hence, the length of one side of the playground is 156 metres.