Question

Please answer this question ASAP

Answer 1

Given Question :- Evaluate

$\rm \: \sqrt[3]{\dfrac{216}{729} } + \sqrt[3]{0.064} + \sqrt[3]{( - 1331) \times (3375)}$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: \sqrt[3]{\dfrac{216}{729} } + \sqrt[3]{0.064} + \sqrt[3]{( - 1331) \times (3375)}$

can be rewritten as

$\rm \: = \sqrt[3]{\dfrac{216}{729} } + \sqrt[3]{\dfrac{64}{1000} } + \sqrt[3]{( - 1331) \times (3375)}$

Now, to evaluate this, let we first convert each term in to its prime factorization.

So,

$\rm \: 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = {6}^{3}$

$\rm \: 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = {9}^{3}$

$\rm \: 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = {4}^{3}$

$\rm \: 1331 = 11 \times 11 \times 11 = {11}^{3}$

$\rm \: 3375 = 3 \times 3 \times 3 \times 5 \times 5 \times 5 = {15}^{3}$

So, the above expression can be rewritten as

$\rm \: = \sqrt[3]{\dfrac{ {6}^{3} }{ {9}^{3} } } + \sqrt[3]{\dfrac{ {4}^{3} }{ {10}^{3} } } + \sqrt[3]{ {( - 11)}^{3} \times {15}^{3} }$

$\rm \: = \: \dfrac{6}{9} + \dfrac{4}{10} + ( - 11)(15)$

$\rm \: = \: \dfrac{2}{3} + \frac{2}{5} - 165$

$\rm \: = \: \dfrac{10 + 6 - 2475}{15}$

$\rm \: = \: -\dfrac{2459}{15}$

Hence,

$\boxed{ \rm{\sqrt[3]{\dfrac{216}{729} } + \sqrt[3]{0.064} + \sqrt[3]{( - 1331) \times (3375)} = - \frac{2459}{15} \: }}$

$\rule{190pt}{2pt}$

Additional information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{ {x}^{0} = 1}\\ \\ \bigstar \: \bf{ {x}^{m} \times {x}^{n} = {x}^{m + n} }\\ \\ \bigstar \: \bf{ {( {x}^{m})}^{n} = {x}^{mn} }\\ \\\bigstar \: \bf{ {x}^{m} \div {x}^{n} = {x}^{m - n} }\\ \\ \bigstar \: \bf{ {x}^{ - n} = \dfrac{1}{ {x}^{n} } }\\ \\\bigstar \: \bf{ {\bigg(\dfrac{a}{b} \bigg) }^{ - n} = {\bigg(\dfrac{b}{a} \bigg) }^{n} }\\ \\\bigstar \: \bf{ {x}^{m} = {x}^{n}\rm\implies \:m = n }\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$

Answer 2

Answer:

$\sf {\blue{ \underline{ \red{Concept:-}}}}$

In the question we have given to evaluate the cube roots. We have to expand the number one by one then find their roots(cube) and then we find the answer.

$\sf{ \red {\underline{To \: \: Evaluate:-}}}$

$\rightarrow \sqrt[3]{ \frac{216}{729} } + \sqrt[3]{0.064} + \sqrt[3]{( - 1331) \times (3375)}$

First we expand them

$\rightarrow \sf \sqrt[3]{ \frac{6 \times 6 \times 6 }{9 \times 9 \times 9} } + \sqrt[3]{0.4 \times 0.4 \times 0.4} + \sqrt[3]{( - 11 \times - 11 \times - 11) \times (15 \times 15 \times 15)}$

Then write in cubic form

$\rightarrow \sf \sqrt[3]{ \frac{( {6})^{3} }{( {9})^{3} } } + \sqrt[3]{( {0.4})^{3} } + \sqrt[3]{ ({ - 11})^{3} \times ( {15})^{3} }$

$\sf \rightarrow { (\frac{6}{9}) }^{3 \times \frac{1}{3} } + ({0.4})^{3 \times \frac{1}{3} } + {( - 11)(15)}^{3 \times \frac{1}{3} }$

Now we cut their cubes,

$\sf \rightarrow \frac{6}{9} + 0.4 + ( - 11 \times 15)$

After removing cubes

$\sf \rightarrow \frac{2}{3} + \frac{4}{10} - 165$

$\sf \rightarrow \frac{2}{3} + \frac{2}{5} - 165$

$\sf \rightarrow \frac{10 + 6 - 2475}{15}$

$\sf \rightarrow \red{ \frac{2459}{15} }$