Question

$\rm \red{Cube \: root \: of \: 0.729 \: is}$​

Answer 1

Answer:

Given Question :-

Find the cube root of 0.729

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

Given expression is

$\rm :\longmapsto\: \sqrt[3]{0.729}$

can be rewritten as

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

Now, we use prime factorization method to find the cube root.

So, Consider

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

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

$\purple{\rm\implies \: Prime \: Factorization \: of \: 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3}$

Now, Consider

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

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

$\green{\rm\implies \:Prime \: Factorization \: of \: 1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5}$

So,

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

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

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

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

$\rm \:  =  \: \dfrac{9}{10}$

$\rm \:  =  \: 0.9$

Hence,

$\rm\implies \:\:\underbrace{\boxed{\tt{ \: \: \: \: \sqrt[3]{0.729} = 0.9 \: \: \: \: }}}$

________________________

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Answer 2

Answer:

Given Question :-

Find the cube root of 0.729

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

Solution−

Given expression is

\rm :\longmapsto\: \sqrt[3]{0.729}:⟼

3

0.729

can be rewritten as

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

3

1000

729

Now, we use prime factorization method to find the cube root.

So, Consider

\purple{\rm :\longmapsto\:Prime \: Factorization \: of \: 729}:⟼PrimeFactorizationof729

\begin{gathered} \purple{\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}}\end{gathered}

3

3

3

3

3

3

729

243

81

27

9

3

1

\purple{\rm\implies \: Prime \: Factorization \: of \: 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3}⟹PrimeFactorizationof729=3×3×3×3×3×3

Now, Consider

\green{\rm :\longmapsto\:Prime \: Factorization \: of \: 1000}:⟼PrimeFactorizationof1000

\begin{gathered} \green{\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}}\end{gathered}

2

2

2

5

5

5

1000

500

250

125

25

5

1

\green{\rm\implies \:Prime \: Factorization \: of \: 1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5}⟹PrimeFactorizationof1000=2×2×2×5×5×5

So,

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

3

1000

729

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

3

2×2×2×5×5×5

3×3×3×3×3×3

\rm \: = \: \sqrt[3]{\dfrac{ \underbrace{3 \times 3 \times 3} \times \underbrace{3 \times 3 \times 3}}{\underbrace{2 \times 2 \times 2} \times \underbrace{5 \times 5 \times 5}} } =

3

2×2×2

×

5×5×5

3×3×3

×

3×3×3

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

2×5

3×3

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

10

9

\rm \: = \: 0.9 = 0.9

Hence,

\rm\implies \:\:\underbrace{\boxed{\tt{ \: \: \: \: \sqrt[3]{0.729} = 0.9 \: \: \: \: }}}⟹

3

0.729

=0.9

________________________

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