Question

$\mathfrak{Question:}$
$\pink{\bold{\underline{\large{ \:↬\ \textless \ br /\ \textgreater \ Find \: the \: cube \: root \: of \: the \: following :}}}} \: \:$
$1. \: \dashrightarrow\ \textless \ br /\ \textgreater \ \: - 1 \frac{271}{729}$
$2.\dashrightarrow\ \textless \ br /\ \textgreater \ \: 2 \frac{93}{125}$
$\hookrightarrow\ \textless \ br /\ \textgreater \ \: Note:\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \:$
Do not spam❌❌
Only mods, stars or quality answerers only answer the question ✅✅​

Answer 1

$\large\underline{\sf{Solution-1}}$

Consider,

$\rm :\longmapsto\: \sqrt[3]{ - \: 1 \: \dfrac{271}{729} }$

We know,

$\blue{ \boxed{\bf \: \sqrt[3]{ - x} = - \sqrt[3]{x}}}$

So, given expression can be rewritten as

$\rm \: = \: \: - \: \sqrt[3]{1 \: \dfrac{271}{729} }$

Now, we reduce this mixed fraction to rational number.

$\rm \: = \: \: - \: \sqrt[3]{\: \dfrac{729 + 271}{729} }$

$\rm \: = \: \: - \: \sqrt[3]{\: \dfrac{1000}{729} }$

$\rm \: = \: - \: \dfrac{ \sqrt[3]{1000} }{ \sqrt[3]{729} }$

Now,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 1000}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:1000 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:500 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:250\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:125 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:25\:\:}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 1000 = {2}^{3} \times {5}^{3} }$

Now, Consider,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 729}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:729 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:243 \:\:}} \\\underline{\sf{3}}&\underline{\sf{\:\:81\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 729 = {3}^{3} \times {3}^{3} }$

Hence,

$\rm :\longmapsto\:\: - \: \dfrac{ \sqrt[3]{1000} }{ \sqrt[3]{729} }$

$\rm \: = \: - \: \dfrac{ \sqrt[3]{ {2}^{3} \times {5}^{3} } }{ \sqrt[3]{ {3}^{3} \times {3}^{3} } }$

$\rm \: = \: \: - \: \dfrac{2 \times 5}{3 \times 3}$

$\rm \: = \: \: - \: \dfrac{10}{9}$

$\rm \: = \: \: - \: 1 \: \dfrac{1}{9}$

$\large\underline{\sf{Solution-2}}$

Consider,

$\rm :\longmapsto\: \sqrt[3]{2 \: \dfrac{93}{125} }$

$\rm \: = \: \: \: \sqrt[3]{\: \dfrac{250 + 93}{125} }$

$\rm \: = \: \: \: \sqrt[3]{\: \dfrac{343}{125} }$

$\rm \: = \: \: \dfrac{ \sqrt[3]{343} }{ \sqrt[3]{125} }$

Now,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 343}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{7}}}&{\underline{\sf{\:\:343 \:\:}}}\\ {\underline{\sf{7}}}& \underline{\sf{\:\:49 \:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:7\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 343 = {7}^{3} }$

Now,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 125 }$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{5}}}&{\underline{\sf{\:\:125 \:\:}}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:25 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 125 = {5}^{3} }$

Hence,

$\rm \: = \: \: \dfrac{ \sqrt[3]{343} }{ \sqrt[3]{125} }$

$\rm \: = \: \: \dfrac{ \sqrt[3]{ {7}^{3} } }{ \sqrt[3]{ {5}^{3} } }$

$\rm \: = \: \: \dfrac{7}{5}$

$\rm \: = \: \:1 \dfrac{2}{5}$

Thus,

$\bf :\longmapsto\:(1). \: \: \sqrt[3]{ - \: 1 \: \dfrac{271}{729} } = - \:1 \: \dfrac{1}{9}$

and

$\bf :\longmapsto\: (2). \: \: \sqrt[3]{2 \: \dfrac{93}{125} } = 1 \: \dfrac{2}{5}$

Result Used :-

$\blue{ \boxed{\bf \: \sqrt[3]{ \frac{x}{y} } \: = \: \frac{ \sqrt[3]{x} }{ \sqrt[3]{y} }}}$