Squares and Square Roots, Cubes and Cube Roots — Exercise 3(D)

Find the square root of 2.89

Answer

Here, the number of decimal places is already even. So, mark the periods and proceed as follows :

Hence, $\mathbf{\sqrt{2.89} = 1.7}$.

Find the square root of 33.64

Answer

Here, the number of decimal places is already even. So, mark the periods and proceed as follows :

Hence, $\mathbf{\sqrt{33.64} = 5.8}$.

Find the square root of 156.25

Answer

Here, the number of decimal places is already even. So, mark the periods and proceed as follows :

Hence, $\mathbf{\sqrt{156.25} = 12.5}$.

Find the square root of 75.69

Answer

Here, the number of decimal places is already even. So, mark the periods and proceed as follows :

Hence, $\mathbf{\sqrt{75.69} = 8.7}$.

Find the square root of 5.4289

Answer

Here, the number of decimal places is already even. So, mark the periods and proceed as follows :

Hence, $\mathbf{\sqrt{5.4289} = 2.33}$.

Find the square root of 18.1476

Answer

Here, the number of decimal places is already even. So, mark the periods and proceed as follows :

Hence, $\mathbf{\sqrt{18.1476} = 4.26}$.

Evaluate $\sqrt{2}$ up to two places of decimal.

Answer

Since the square root is required to be correct to 2 places of decimal, we shall find it up to 3 places of decimal and round it off to 2 places of decimal.

For finding the square root up to 3 decimal places, we need 3 pairs of decimal places.

So, we write 2 = 2.000000.

Now, mark off periods and extract the square root as shown below :

$\sqrt{2} = 1.414$ (up to 3 places of decimal)

$\sqrt{2} = 1.41$ (correct up to 2 places of decimal)

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

Evaluate $\sqrt{0.4}$ correct up to two places of decimal.

Answer

Since the square root is required to be correct to 2 places of decimal, we shall find it up to 3 places of decimal and round it off to 2 places of decimal.

So, we write 0.4 = 0.400000.

Now, mark off periods and extract the square root as shown below :

$\sqrt{0.4} = 0.632$ (up to 3 places of decimal)

$\sqrt{0.4} = 0.63$ (correct up to 2 places of decimal)

Hence, $\mathbf{\sqrt{0.4} = 0.63}$.

Evaluate $\sqrt{2.8}$ correct up to two places of decimal.

Answer

Since the square root is required to be correct to 2 places of decimal, we shall find it up to 3 places of decimal and round it off to 2 places of decimal.

So, we write 2.8 = 2.800000.

Now, mark off periods and extract the square root as shown below :

$\sqrt{2.8} = 1.673$ (up to 3 places of decimal)

$\sqrt{2.8} = 1.67$ (correct up to 2 places of decimal)

Hence, $\mathbf{\sqrt{2.8} = 1.67}$.

Find the length of each side of a square whose area is equal to the area of a rectangle of length 13.6 metres and breadth 3.4 metres.

Answer

By formula,

Area of the rectangle = length × breadth

Area of the rectangle = 13.6 x 3.4

Area of the rectangle = 46.24 m²

Let the length of each side of the square be $x$ metres.

Area of the square = Area of the rectangle

$\Rightarrow x^2 = 46.24\\[1em] \Rightarrow x = \sqrt{46.24}$

Marking periods and using the division method, we have :

⇒ $x = 6.8$

Hence, the length of each side of the square is 6.8 metres.

If $\sqrt{1 + \dfrac{27}{169}} = 1 + \dfrac{x}{13}$, find the value of $x$.

Answer

Solving,

$\Rightarrow \sqrt{1 + \dfrac{27}{169}} = 1 + \dfrac{x}{13}\\[1em] \Rightarrow \sqrt{\dfrac{169 + 27}{169}} = 1 + \dfrac{x}{13}\\[1em] \Rightarrow \sqrt{\dfrac{196}{169}} = 1 + \dfrac{x}{13}\\[1em] \Rightarrow \dfrac{\sqrt{196}}{\sqrt{169}} = 1 + \dfrac{x}{13}\\[1em] \Rightarrow \dfrac{14}{13} = 1 + \dfrac{x}{13}\\[1em] \Rightarrow 1 + \dfrac{1}{13} = 1 + \dfrac{x}{13}\\[1em] \Rightarrow \dfrac{1}{13} = \dfrac{x}{13}\\[1em] \Rightarrow x = 1$

Hence, $\bm{x = 1}$.

Find the square root of:

$3\dfrac{13}{36}$

Answer

Solving,

$\Rightarrow \sqrt{3\dfrac{13}{36}}\\[1em] \Rightarrow \sqrt{\dfrac{121}{36}}\\[1em] \Rightarrow \dfrac{\sqrt{121}}{\sqrt{36}}\\[1em] \Rightarrow \dfrac{\sqrt{11 \times 11}}{\sqrt{(2 \times 2) \times (3 \times 3)}}\\[1em] \Rightarrow \dfrac{11}{6}\\[1em] \Rightarrow 1\dfrac{5}{6}$

Hence, $\mathbf{\sqrt{3\dfrac{13}{36}} = 1\dfrac{5}{6}}$.

Find the square root of:

$4\dfrac{73}{324}$

Answer

Solving,

$\Rightarrow \sqrt{4\dfrac{73}{324}}\\[1em] \Rightarrow \sqrt{\dfrac{1369}{324}}\\[1em] \Rightarrow \dfrac{\sqrt{1369}}{\sqrt{324}}\\[1em] \Rightarrow \dfrac{\sqrt{37 \times 37}}{\sqrt{(2 \times 2) \times (3 \times 3) \times (3 \times 3)}}\\[1em] \Rightarrow \dfrac{37}{18}\\[1em] \Rightarrow 2\dfrac{1}{18}$

Hence, $\mathbf{\sqrt{4\dfrac{73}{324}} = 2\dfrac{1}{18}}$.

Find the square root of:

$3\dfrac{942}{2209}$

Answer

Solving,

$\Rightarrow \sqrt{3\dfrac{942}{2209}}\\[1em] \Rightarrow \sqrt{\dfrac{7569}{2209}}\\[1em] \Rightarrow \dfrac{\sqrt{7569}}{\sqrt{2209}}\\[1em] \Rightarrow \dfrac{\sqrt{(3 \times 3) \times (29 \times 29)}}{\sqrt{47 \times 47}}\\[1em] \Rightarrow \dfrac{3 \times 29}{47}\\[1em] \Rightarrow \dfrac{87}{47}\\[1em] \Rightarrow 1\dfrac{40}{47}$

Hence, $\mathbf{\sqrt{3\dfrac{942}{2209}} = 1\dfrac{40}{47}}$.

Find the square root of:

$\dfrac{1089}{4624}$

Answer

Solving,

$\Rightarrow \sqrt{\dfrac{1089}{4624}}\\[1em] \Rightarrow \dfrac{\sqrt{1089}}{\sqrt{4624}}\\[1em] \Rightarrow \dfrac{\sqrt{(3 \times 3) \times (11 \times 11)}}{\sqrt{(2 \times 2) \times (2 \times 2) \times (17 \times 17)}}\\[1em] \Rightarrow \dfrac{3 \times 11}{2 \times 2 \times 17}\\[1em] \Rightarrow \dfrac{33}{68}$

Hence, $\mathbf{\sqrt{\dfrac{1089}{4624}} = \dfrac{33}{68}}$.

Find the value of:

$\dfrac{\sqrt{243}}{\sqrt{867}}$

Answer

Solving,

$\Rightarrow \dfrac{\sqrt{243}}{\sqrt{867}}\\[1em] \Rightarrow \sqrt{\dfrac{243}{867}}\\[1em] \Rightarrow \sqrt{\dfrac{\cancel{3} \times 81}{\cancel{3} \times 289}}\\[1em] \Rightarrow \dfrac{\sqrt{81}}{\sqrt{289}}\\[1em] \Rightarrow \dfrac{\sqrt{9 \times 9}}{\sqrt{17 \times 17}}\\[1em] \Rightarrow \dfrac{9}{17}$

Hence, $\mathbf{\dfrac{\sqrt{243}}{\sqrt{867}} = \dfrac{9}{17}}$.

Find the value of:

$\dfrac{\sqrt{1183}}{\sqrt{2023}}$

Answer

Solving,

$\Rightarrow \dfrac{\sqrt{1183}}{\sqrt{2023}}\\[1em] \Rightarrow \sqrt{\dfrac{1183}{2023}}\\[1em] \Rightarrow \sqrt{\dfrac{\cancel{7} \times 169}{\cancel{7} \times 289}}\\[1em] \Rightarrow \sqrt{\dfrac{169}{289}}\\[1em] \Rightarrow \dfrac{\sqrt{169}}{\sqrt{289}}\\[1em] \Rightarrow \dfrac{\sqrt{13 \times 13}}{\sqrt{17 \times 17}}\\[1em] \Rightarrow \dfrac{13}{17}$

Hence, $\mathbf{\dfrac{\sqrt{1183}}{\sqrt{2023}} = \dfrac{13}{17}}$.

Find the value of $\sqrt{15625}$ and hence evaluate $\sqrt{156.25} + \sqrt{1.5625}$.

Answer

Marking periods and using the division method, we have :

∴ $\sqrt{15625} = 125$

Now, evaluating $\sqrt{156.25} + \sqrt{1.5625}$,

$\Rightarrow \sqrt{156.25} + \sqrt{1.5625}\\[1em] \Rightarrow \sqrt{\dfrac{15625}{100}} + \sqrt{\dfrac{15625}{10000}}\\[1em] \Rightarrow \dfrac{\sqrt{15625}}{\sqrt{100}} + \dfrac{\sqrt{15625}}{\sqrt{10000}}\\[1em] \Rightarrow \dfrac{125}{10} + \dfrac{125}{100}\\[1em] \Rightarrow 12.5 + 1.25\\[1em] \Rightarrow 13.75$

Hence, $\mathbf{\sqrt{15625} = 125}$ and $\mathbf{\sqrt{156.25} + \sqrt{1.5625} = 13.75}$.

Evaluate:

$\sqrt{99} \times \sqrt{396}$

Answer

Evaluating,

$\Rightarrow \sqrt{99} \times \sqrt{396}\\[1em] \Rightarrow \sqrt{99 \times 396}\\[1em] \Rightarrow \sqrt{(3 \times 3) \times 11 \times (2 \times 2) \times (3 \times 3) \times 11}\\[1em] \Rightarrow \sqrt{(2 \times 2) \times (3 \times 3) \times (3 \times 3) \times (11 \times 11)}\\[1em] \Rightarrow 2 \times 3 \times 3 \times 11\\[1em] \Rightarrow 198$

Hence, $\mathbf{\sqrt{99} \times \sqrt{396} = 198}$.

Evaluate:

$\sqrt{147} \times \sqrt{243}$

Answer

Evaluating,

$\Rightarrow \sqrt{147} \times \sqrt{243}\\[1em] \Rightarrow \sqrt{147 \times 243}\\[1em] \Rightarrow \sqrt{3 \times (7 \times 7) \times (3 \times 3) \times (3 \times 3) \times 3}\\[1em] \Rightarrow \sqrt{(3 \times 3) \times (3 \times 3) \times (3 \times 3) \times (7 \times 7)}\\[1em] \Rightarrow 3 \times 3 \times 3 \times 7\\[1em] \Rightarrow 189$

Hence, $\mathbf{\sqrt{147} \times \sqrt{243} = 189}$.

Evaluate:

$\sqrt{\dfrac{0.289}{0.00121}}$

Answer

Converting the decimals into a fraction of whole numbers:

$\Rightarrow \sqrt{\dfrac{0.289}{0.00121}}\\[1em] \Rightarrow \sqrt{\dfrac{28900}{121}}\\[1em] \Rightarrow \dfrac{\sqrt{28900}}{\sqrt{121}}\\[1em] \Rightarrow \dfrac{\sqrt{(2 \times 2) \times (5 \times 5) \times (17 \times 17)}}{\sqrt{11 \times 11}}\\[1em] \Rightarrow \dfrac{2 \times 5 \times 17}{11}\\[1em] \Rightarrow \dfrac{170}{11}\\[1em] \Rightarrow 15\dfrac{5}{11}$

Hence, $\mathbf{\sqrt{\dfrac{0.289}{0.00121}} = \dfrac{170}{11} = 15\dfrac{5}{11}}$.

Evaluate:

$\sqrt{\dfrac{48.4}{0.289}}$

Answer

Converting the decimals into a fraction of whole numbers:

$\Rightarrow \sqrt{\dfrac{48.4}{0.289}}\\[1em] \Rightarrow \sqrt{\dfrac{48400}{289}}\\[1em] \Rightarrow \dfrac{\sqrt{48400}}{\sqrt{289}}\\[1em] \Rightarrow \dfrac{\sqrt{(2 \times 2) \times (5 \times 5) \times (11 \times 11) \times (2 \times 2)}}{\sqrt{17 \times 17}}\\[1em] \Rightarrow \dfrac{2 \times 5 \times 11 \times 2}{17}\\[1em] \Rightarrow \dfrac{220}{17}\\[1em] \Rightarrow 12\dfrac{16}{17}$

Hence, $\mathbf{\sqrt{\dfrac{48.4}{0.289}} = \dfrac{220}{17} = 12\dfrac{16}{17}}$.

Evaluate:

$\sqrt{0.01 + \sqrt{0.0064}}$

Answer

Evaluating,

$\Rightarrow \sqrt{0.01 + \sqrt{0.0064}}\\[1em] [\because \sqrt{0.0064} = 0.08]\\[1em] \Rightarrow \sqrt{0.01 + 0.08}\\[1em] \Rightarrow \sqrt{0.09}\\[1em] \Rightarrow 0.3$

Hence, $\mathbf{\sqrt{0.01 + \sqrt{0.0064}} = 0.3}$.

Evaluate:

$\sqrt{0.01} + \sqrt{0.81} + \sqrt{1.21} + \sqrt{0.0009}$

Answer

Evaluating,

⇒ $\sqrt{0.01} + \sqrt{0.81} + \sqrt{1.21} + \sqrt{0.0009}$

⇒ 0.1 + 0.9 + 1.1 + 0.03

⇒ 2.13

Hence, $\mathbf{\sqrt{0.01} + \sqrt{0.81} + \sqrt{1.21} + \sqrt{0.0009} = 2.13}$.

Evaluate:

$\sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}}$

Answer

Evaluating,

$\Rightarrow \sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}}\\[1em] [\because \sqrt{9} = 3]\\[1em] \Rightarrow \sqrt{41 - \sqrt{21 + \sqrt{19 - 3}}}\\[1em] \Rightarrow \sqrt{41 - \sqrt{21 + \sqrt{16}}}\\[1em] [\because \sqrt{16} = 4]\\[1em] \Rightarrow \sqrt{41 - \sqrt{21 + 4}}\\[1em] \Rightarrow \sqrt{41 - \sqrt{25}}\\[1em] [\because \sqrt{25} = 5]\\[1em] \Rightarrow \sqrt{41 - 5}\\[1em] \Rightarrow \sqrt{36}\\[1em] \Rightarrow 6$

Hence, $\mathbf{\sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}} = 6}$.

Evaluate:

$\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}}$

Answer

Evaluating,

$\Rightarrow \sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}}\\[1em] [\because \sqrt{225} = 15]\\[1em] \Rightarrow \sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + 15}}}}\\[1em] \Rightarrow \sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{169}}}}\\[1em] [\because \sqrt{169} = 13]\\[1em] \Rightarrow \sqrt{10 + \sqrt{25 + \sqrt{108 + 13}}}\\[1em] \Rightarrow \sqrt{10 + \sqrt{25 + \sqrt{121}}}\\[1em] [\because \sqrt{121} = 11]\\[1em] \Rightarrow \sqrt{10 + \sqrt{25 + 11}}\\[1em] \Rightarrow \sqrt{10 + \sqrt{36}}\\[1em] [\because \sqrt{36} = 6]\\[1em] \Rightarrow \sqrt{10 + 6}\\[1em] \Rightarrow \sqrt{16}\\[1em] \Rightarrow 4$

Hence, $\mathbf{\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}} = 4}$.

Three-fifths of the square of a certain number is 126.15. Find the number.

Answer

Let the number be $x$.

Three-fifths of the square of the number = 126.15

$\Rightarrow \dfrac{3}{5}x^2 = 126.15\\[1em] \Rightarrow x^2 = \dfrac{126.15 \times 5}{3}\\[1em] \Rightarrow x^2 = \dfrac{630.75}{3}\\[1em] \Rightarrow x^2 = 210.25\\[1em] \Rightarrow x = \sqrt{210.25}$

Marking periods and using the division method, we have :

⇒ $x = 14.5$

Hence, the required number is 14.5.