Square And Qube

$${\text{Let ,}}$$
$$\sqrt {\frac{{0.0196}}{x}} = 0.2$$
$${\text{Then,}}$$
$$ ⇒ \frac{{0.0196}}{x} = 0.04$$
$$ ⇒ x = \frac{{0.0196}}{{0.04}}$$
$$ ⇒ x = \frac{{1.96}}{4}$$
$$ ⇒ x = 0.49$$

$$ ⇒ 37 + \sqrt {0.0615 + x} = 37.25$$
$$ ⇒ \sqrt {0.0615 + x} = 0.25$$
$$ ⇒ 0.0615 +×= {\left( {0.25} \right)^2} = 0.0625$$
$$ ⇒ x = 0.001$$
$$ ⇒ x = \frac{1}{{{{10}^3}}}$$
$$ ⇒ x = {10^{ - 3}}$$

$$ ⇒ \sqrt {\left( {x - 1} \right)\left( {y + 2} \right)} = 7$$
$$ ⇒ \left( {x - 1} \right)\left( {y + 2} \right) = {\left( 7 \right)^2}$$
$$ ⇒ \left( {x - 1} \right) = 7{\text{ and}}\left( {y + 2} \right) = 7$$
$$ ⇒ x = 8{\text{ and }}y = 5$$

$${\text{Let the number be }}x$$
$${\text{Then,}}$$
$$ ⇒ \frac{3}{5}{x^2} = 126.15$$
$$ ⇒ {x^2} = \left( {126.15 \times \frac{5}{3}} \right)$$
$$ ⇒ {x^2} = 210.25$$
$$ ⇒ x = \sqrt {210.25} $$
$$ ⇒ x = 14.5$$

$$ ⇒ \sqrt {1 + \frac{x}{{169}}} = \frac{{14}}{{13}}$$
$$ ⇒ 1 + \frac{x}{{169}} = \frac{{196}}{{169}}$$
$$ ⇒ \frac{x}{{169}} = \left( {\frac{{196}}{{169}} - 1} \right)$$
$$ ⇒ \frac{x}{{169}} = \frac{{27}}{{169}}$$
$$ ⇒ x = 27$$

Sum of decimal places in the numerator and denominator under the radical sign being the same, we remove the decimal.
∴ Given expression,
$$ = \sqrt {\frac{{0.081 \times 0.484}}{{0.0064 \times 6.25}}} $$
$$ = \sqrt {\frac{{81 \times 484}}{{64 \times 625}}} $$
$$ = \frac{{9 \times 22}}{{8 \times 25}}$$
$$ = 0.99$$

$${\text{Given expression,}}$$
$$ = \sqrt {\frac{{{{\left( {0.03} \right)}^2} + {{\left( {0.21} \right)}^2} + {{\left( {0.065} \right)}^2}}}{{{{\left( {\frac{{0.03}}{{10}}} \right)}^2} + {{\left( {\frac{{0.21}}{{10}}} \right)}^2} + {{\left( {\frac{{0.065}}{{10}}} \right)}^2}}}} $$
$$ = \sqrt {\frac{{100\left[ {{{\left( {0.03} \right)}^2} + {{\left( {0.21} \right)}^2} + {{\left( {0.065} \right)}^2}} \right]}}{{{{\left( {0.03} \right)}^2} + {{\left( {0.21} \right)}^2} + {{\left( {0.065} \right)}^2}}}} $$
$$ = \sqrt {100} $$
$$ = 10$$

$$ = \sqrt {\left( {7 + 3\sqrt 5 } \right)\left( {7 - 3\sqrt 5 } \right)} $$
$$ = \sqrt {{{\left( 7 \right)}^2} - {{\left( {3\sqrt 5 } \right)}^2}} $$
$$ = \sqrt {49 - 45} $$
$$ = \sqrt 4 $$
$$ = 2$$

$$ = {\left( {\sqrt 3 - \frac{1}{{\sqrt 3 }}} \right)^2}$$
$$ = {\left( {\sqrt 3 } \right)^2} + {\left( {\frac{1}{{\sqrt 3 }}} \right)^2} - 2 \times \sqrt 3 \times \frac{1}{{\sqrt 3 }}$$
$$ = 3 + \frac{1}{3} - 2$$
$$ = 1 + \frac{1}{3}$$
$$ = \frac{4}{3}$$

$$ = \sqrt {4{a^2} - 4a + 1} + 3a$$
$$ = \sqrt {{{\left( 1 \right)}^2} + {{\left( {2a} \right)}^2} - 2 \times 1 \times 2a} + 3a$$
$$ = \sqrt {{{\left( {1 - 2a} \right)}^2}} + 3a$$
$$ = \left( {1 - 2a} \right) + 3a$$
$$ = \left( {1 + a} \right)$$
$$ = \left( {1 + 0.1039} \right)$$
$$ = 1.1039$$