Question

# 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) $10\frac{151}{225}$

Solution

## (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. $\sqrt{441}=\sqrt{\left(3×3\right)×\left(7×7\right)}=3×7=21\phantom{\rule{0ex}{0ex}}\sqrt{961}=\sqrt{31×31}=31\phantom{\rule{0ex}{0ex}}\therefore \sqrt{\frac{441}{961}}=\frac{21}{31}$ (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. $\sqrt{324}=\sqrt{2×2×3×3×3×3}=2×3×3=18\phantom{\rule{0ex}{0ex}}\sqrt{841}=\sqrt{29×29}=29\phantom{\rule{0ex}{0ex}}\therefore \sqrt{\frac{324}{841}}=\frac{18}{29}$ (iii) By looking at the book's answer key, the fraction should be . We know: $\sqrt{4\frac{29}{49}}=\sqrt{\frac{225}{49}}=\frac{\sqrt{225}}{\sqrt{49}}\phantom{\rule{0ex}{0ex}}\sqrt{225}=15\phantom{\rule{0ex}{0ex}}\sqrt{49}=7\phantom{\rule{0ex}{0ex}}\therefore \sqrt{4\frac{29}{49}}=\frac{15}{7}\phantom{\rule{0ex}{0ex}}$ (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. $\sqrt{529}=\sqrt{23×23}=23\phantom{\rule{0ex}{0ex}}\sqrt{196}=\sqrt{2×2×7×7}=2×7=14\phantom{\rule{0ex}{0ex}}\therefore \sqrt{2\frac{137}{196}}=\frac{23}{14}$ (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. $\sqrt{121}=11\phantom{\rule{0ex}{0ex}}\therefore \sqrt{23\frac{26}{121}}=\frac{53}{11}$ (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. $\sqrt{729}=27\phantom{\rule{0ex}{0ex}}\therefore \sqrt{25\frac{544}{729}}=\frac{137}{27}$ (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. $\sqrt{49}=7\phantom{\rule{0ex}{0ex}}\therefore \sqrt{75\frac{46}{49}}=\frac{61}{7}$ (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. $\sqrt{961}=31\phantom{\rule{0ex}{0ex}}\sqrt{25}=5\phantom{\rule{0ex}{0ex}}\therefore \sqrt{38\frac{11}{25}}=\frac{31}{5}$ (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. $\sqrt{729}=27\phantom{\rule{0ex}{0ex}}\therefore \sqrt{23\frac{394}{729}}=\frac{131}{27}=4\frac{23}{27}$ (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. $\sqrt{3600}=\sqrt{60×60}=60\phantom{\rule{0ex}{0ex}}\sqrt{169}=\sqrt{13×13}=13\phantom{\rule{0ex}{0ex}}\therefore \sqrt{21\frac{51}{169}}=\frac{60}{13}=4\frac{8}{13}\phantom{\rule{0ex}{0ex}}$ (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. $\sqrt{2401}=\sqrt{7×7×7×7}=7×7=49\phantom{\rule{0ex}{0ex}}\sqrt{225}=\sqrt{3×3×5×5}=3×5=15\phantom{\rule{0ex}{0ex}}\therefore \sqrt{10\frac{151}{225}}=\frac{49}{15}=3\frac{4}{15}$MathematicsRD Sharma (2019, 2020)All

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