Find the square root of:
(i) \( \frac{441}{961} \)
(ii) \( \frac{324}{841} \)
(iii) \( 4 \frac{29}{49} \)
(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} \)

To do:

We have to find the square root of the given number.

Solution:

(i) $\sqrt{\frac{441}{961}}=\frac{\sqrt{441}}{\sqrt{961}}$

Square root of 441 is,

Square root of 961 is,

Therefore,

$\sqrt{\frac{441}{961}}=\frac{21}{31}$.

(ii) $\sqrt{\frac{324}{841}}=\frac{\sqrt{324}}{\sqrt{841}}$

Square root of 324 is,

Square root of 841 is,

Therefore,

$\sqrt{\frac{324}{841}}=\frac{18}{29}$.

(iii) $4 \frac{29}{49}=\frac{4\times49+29}{49}$

$=\frac{196+29}{49}$

$=\frac{225}{49}$

$\sqrt{\frac{225}{49}}=\frac{\sqrt{225}}{\sqrt{49}}$

Square root of 225 is,

Square root of 49 is 7.

Therefore,

$\sqrt{4\frac{29}{49}}=\frac{15}{7}$.

(iv) $2 \frac{14}{25}=\frac{2\times25+14}{25}$

$=\frac{50+14}{25}$

$=\frac{64}{25}$

$\sqrt{\frac{64}{25}}=\frac{\sqrt{64}}{\sqrt{25}}$

Square root of 64 is 8.

Square root of 25 is 5.

Therefore,

$\sqrt{2\frac{14}{25}}=\frac{8}{5}$. 

(v) $2 \frac{137}{196}=\frac{2\times196+137}{196}$

$=\frac{392+137}{196}$

$=\frac{529}{196}$

$\sqrt{\frac{529}{196}}=\frac{\sqrt{529}}{\sqrt{196}}$

Square root of 529 is 23.

Square root of 196 is 14.

Therefore,

$\sqrt{2\frac{137}{196}}=\frac{23}{14}$. 

(vi) $23 \frac{26}{121}=\frac{23\times121+26}{121}$

$=\frac{2783+26}{121}$

$=\frac{2809}{121}$

$\sqrt{\frac{2809}{121}}=\frac{\sqrt{2809}}{\sqrt{121}}$

Square root of 2809 is,

Square root of 2809 is 53.

Square root of 121 is,

Square root of 121 is 11.

Therefore,

$\sqrt{23\frac{26}{121}}=\frac{53}{11}$.

(vii) $25 \frac{544}{729}=\frac{25\times729+544}{729}$

$=\frac{18225+544}{729}$

$=\frac{18769}{729}$

$\sqrt{\frac{18769}{729}}=\frac{\sqrt{18769}}{\sqrt{729}}$

Square root of 18769 is,

Square root of 18769 is 137.

Square root of 729 is,

Square root of 729 is 27.

Therefore,

$\sqrt{25\frac{544}{729}}=\frac{137}{27}$.

(viii) $75 \frac{46}{49}=\frac{75\times49+46}{49}$

$=\frac{3675+46}{49}$

$=\frac{3721}{49}$

$\sqrt{\frac{3721}{49}}=\frac{\sqrt{3721}}{\sqrt{49}}$

Square root of 3721 is,

Square root of 3721 is 61.

Square root of 49 is 7.

Therefore,

$\sqrt{75\frac{46}{49}}=\frac{61}{7}$.

(ix) $3 \frac{942}{2209}=\frac{3\times2209+942}{2209}$

$=\frac{6627+942}{2209}$

$=\frac{7569}{2209}$

$\sqrt{\frac{7569}{2209}}=\frac{\sqrt{7569}}{\sqrt{2209}}$

Square root of 7569 is,

Square root of 7569 is 87.

Square root of 2209 is,

Square root of 2209 is 47.

Therefore,

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

(x) $3 \frac{334}{3025}=\frac{3\times3025+334}{3025}$

$=\frac{9075+334}{3025}$

$=\frac{9409}{3025}$

$\sqrt{\frac{9409}{3025}}=\frac{\sqrt{9409}}{\sqrt{3025}}$

Square root of 9409 is,

Square root of 9409 is 97.

Square root of 3025 is,

Square root of 3025 is 55.

Therefore,

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

(xi) $21 \frac{2797}{3364}=\frac{21\times3364+2797}{3364}$

$=\frac{70644+2797}{3364}$

$=\frac{73441}{3364}$

$\sqrt{\frac{73441}{3364}}=\frac{\sqrt{73441}}{\sqrt{3364}}$

Square root of 73441 is,

Square root of 73441 is 271.

Square root of 3364 is,

Square root of 3364 is 58.

Therefore,

$\sqrt{21\frac{2797}{3364}}=\frac{271}{58}$.$21 \frac{2797}{3364}=\frac{21\times3364+2797}{3364}$

$=\frac{70644+2797}{3364}$

$=\frac{73441}{3364}$

$\sqrt{\frac{73441}{3364}}=\frac{\sqrt{73441}}{\sqrt{3364}}$

Square root of 73441 is,

Square root of 73441 is 271.

Square root of 3364 is,

Square root of 3364 is 58.

Therefore,

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

(xii) $38 \frac{11}{25}=\frac{38\times25+11}{25}$

$=\frac{950+11}{25}$

$=\frac{961}{25}$

$\sqrt{\frac{961}{25}}=\frac{\sqrt{961}}{\sqrt{25}}$

Square root of 961 is,

Square root of 961 is 31.

Square root of 25 is 5.

Therefore,

$\sqrt{38\frac{11}{25}}=\frac{31}{5}$. 

(xiii) $23 \frac{394}{729}=\frac{23\times729+394}{729}$

$=\frac{16767+394}{729}$

$=\frac{17161}{729}$

$\sqrt{\frac{17161}{729}}=\frac{\sqrt{17161}}{\sqrt{729}}$

Square root of 17161 is,

Square root of 17161 is 131.

Square root of 729 is,

Square root of 729 is 27.

Therefore,

$\sqrt{23\frac{394}{729}}=\frac{131}{27}$.

(xiv) $21 \frac{51}{169}=\frac{21\times169+51}{169}$

$=\frac{3549+51}{169}$

$=\frac{3600}{169}$

$\sqrt{\frac{3600}{169}}=\frac{\sqrt{3600}}{\sqrt{169}}$

Square root of 3600 is 60.

Square root of 169 is 13.

Therefore,

$\sqrt{21\frac{51}{169}}=\frac{60}{13}$. 

(xv) $10 \frac{151}{225}=\frac{10\times225+151}{225}$

$=\frac{2250+151}{225}$

$=\frac{2401}{225}$

$\sqrt{\frac{2401}{225}}=\frac{\sqrt{2401}}{\sqrt{225}}$

Square root of 2401 is,

Square root of 2401 is 49.

Square root of 225 is,

Square root of 225 is 15.

Therefore,

$\sqrt{10\frac{151}{225}}=\frac{49}{15}$.

Tutorialspoint

Updated on: 10-Oct-2022

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