Simplification

$${\text{we know that}}$$
$${\text{ }}\frac{{{{\left( {3.4567} \right)}^2} - {{\left( {3.4533} \right)}^2}}}{{0.0034}}$$
$$\left[ {\because {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)} \right]$$
$$ ⇒ \frac{{\left( {3.4567 + 3.4533} \right)\left( {3.4567 - 3.4533} \right)}}{{0.0034}}$$
$$ ⇒ \frac{{6.91 \times 0.0034}}{{0.0034}}$$
$$ ⇒ 6.91$$

$${\text{we know that,}}$$
$$\frac{{{{\left( {0.75} \right)}^3}}}{{1 - 0.75}} + \left[ {0.75 + {{\left( {0.75} \right)}^2} + 1} \right]$$
$$ ⇒ \frac{{{{\left( {0.75} \right)}^3}}}{{1 - 0.75}} + \frac{{{1^3} - {{\left( {0.75} \right)}^3}}}{{1 - 0.75}}\,$$
$$\left[ {\because {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)} \right]$$
$$ ⇒ \frac{{{{\left( {0.75} \right)}^3} + 1 - {{\left( {0.75} \right)}^3}}}{{0.25}}$$
$$ ⇒ \frac{1}{{0.25}}$$
$$ ⇒ 4$$

$${\text{we know that,}}$$
$$\sqrt {4 + \sqrt {44 + \sqrt {10000} } } $$
$$ ⇒ \sqrt {4 + \sqrt {44 + 100} } $$
$$ ⇒ \sqrt {4 + \sqrt {144} } $$
$$ ⇒ \sqrt {4 + 12} $$
$$ ⇒ \sqrt {16} $$
$$ ⇒ 4$$

$${\text{Let,}}$$
$$45 - \left[ {28 - \left\{ {37 - \left( {15 - x} \right)} \right\}} \right] = 58$$
$${\text{then,}}$$
$${\text{45}} - \left[ {28 - \left\{ {37 - 15 + x} \right\}} \right] = 58$$
$$ ⇒ 45 - \left[ {28 - \left\{ {22 + x} \right\}} \right] = 58$$
$$ ⇒ 45 - \left[ {28 - 22 - x} \right] = 58$$
$$ ⇒ 45 - \left[ {6 - x} \right] = 58$$
$$ ⇒ 45 - 6 +×= 58$$
$$ ⇒ 39 +×= 58$$
$$ ⇒ x = 58 - 39 $$
$$ ⇒ x = 19$$

$$\frac{{113 \times 4 -×\times 2}}{{13 \times 9 - 5 \times 7}} = 5$$
$$ ⇒ \frac{{452 - 2x}}{{117 - 35}} = 5$$
$$ ⇒ \frac{{452 - 2x}}{{82}} = 5$$
$$ ⇒ 452 - 2x = 410$$
$$ ⇒ 2x = 452 - 410 = 42$$
$$ ⇒ x = 21$$

( 2² + 3² + 4² + .......... + 20² )
= 2² ( 1² + 2² + 3² + .......... + 10² )
= 4 × 385
= 1540

$${\text{we know that,}}$$
$$\frac{{\root 3 \of 8 }}{{\sqrt {16} }} \div \sqrt {\frac{{100}}{{49}}} \times \root 3 \of {125} \,$$
$$ ⇒ \frac{2}{4} \div \frac{{10}}{7} \times 5$$
$$ ⇒ \frac{2}{4} \times \frac{7}{{10}} \times 5$$
$$ ⇒ \frac{7}{4}$$
$$ ⇒ 1\frac{3}{4}$$

$${\text{we know that,}}$$
$$\frac{{\frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6}}}{{\frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{{18}}}}$$
$$ ⇒ \frac{{\frac{{30 - 15 + 12 - 10}}{{60}}}}{{\frac{{36 - 50 + 54 - 35}}{{90}}}}$$
$$ ⇒ \frac{{17}}{{60}} \times \frac{{90}}{5}$$
$$ ⇒ 5\frac{1}{{10}}$$

$${\text{we know that,}}$$
$$\frac{{\sqrt {72} - \sqrt {18} }}{{\sqrt {12} }}$$
$$ ⇒ \sqrt {\frac{{72}}{{12}}} - \sqrt {\frac{{18}}{{12}}} $$
$$ ⇒ \sqrt 6 - \sqrt {\frac{6}{4}} $$
$$ ⇒ \frac{{\sqrt 6 }}{2}$$

$${\text{Given expression,}}$$
$$\frac{7}{4} - \frac{6}{5} + \frac{{13}}{8}$$
$$ = \frac{{70 - 48 + 65}}{{40}}$$
$$ = \frac{{87}}{{40}}$$
$$ = 2\frac{7}{{40}}$$