Question

give the answer in step by step only ​

Answer 1

Step-by-step explanation:

I hope you got this answer

Answer 2

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

Given Question is

$\rm :\longmapsto\: \sqrt[3]{4\dfrac{508}{1331} }$

Let we first convert this mixed fraction to simplest fraction form.

$\rm  \:  =  \: \: \sqrt[3]{\dfrac{1331 \times 4 + 508}{1331} }$

$\rm  \:  =  \: \: \sqrt[3]{\dfrac{5324 + 508}{1331} }$

$\rm  \:  =  \: \: \sqrt[3]{\dfrac{5832}{1331} }$

$\rm  \:  =  \: \:\dfrac{ \sqrt[3]{5832} }{ \sqrt[3]{1331} }$

$\red{\bigg \{ \because \: \sqrt[3]{\dfrac{x}{y} } = \dfrac{ \sqrt[3]{x} }{ \sqrt[3]{y} }\bigg \}}$

Let now find the prime factorization of 5832 and 1331.

Consider,

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 1331}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{11}}}&{\underline{\sf{\:\:1331 \:\:}}}\\ {\underline{\sf{11}}}& \underline{\sf{\:\:121\:\:}} \\\underline{\sf{11}}&\underline{\sf{\:\:11 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 1331 = 11 \times 11 \times 11}$

$\rm :\longmapsto\: \sqrt[3]{1331}$

$\rm  \:  =  \: \: \sqrt[3]{ \underbrace{11 \times 11 \times 11}}$

$\rm  \:  =  \: \:11$

$\bf\implies \: \sqrt[3]{1331} = 11$

Now,

Consider,

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 5832}$

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:5832 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:2916 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:1458\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:729 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:243 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:81 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 5832}$

$\rm  \:  =  \: \:2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$

Hence,

$\rm :\longmapsto\: \sqrt[3]{5832}$

$\rm  \:  =  \: \: \sqrt[3]{ \underbrace{2 \times 2 \times 2} \times \underbrace{3 \times 3 \times 3} \times \underbrace{3 \times 3 \times 3}}$

$\rm  \:  =  \: \:2 \times 3 \times 3$

$\rm  \:  =  \: \:18$

$\bf :\implies\: \sqrt[3]{5832} = 18$

Therefore,

$\rm :\longmapsto\:\dfrac{ \sqrt[3]{5832} }{ \sqrt[3]{1331} } = \dfrac{18}{11} = 1 \: \dfrac{7}{11}$

Hence,

$\purple{\bf :\longmapsto\: \sqrt[3]{4\dfrac{508}{1331} } = 1 \: \dfrac{7}{11} }$

Additional Information :-

1. Each prime factor appear thrice in its cube root if its a perfect cube.

2. Cubes of all even number is even.

3. Cube of all odd number is odd.

4. Cube root of a negative perfect cube is negative.

5. Method of finding cube roots

There are two methods to find the cube root

1. Prime factorization method

2. Using unit digit method