[tex]\large\underline{\sf{Solution-1}}[/tex]
Given expression is
[tex]\sf \: \sqrt[3]{64 \times 729} \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \sqrt[3]{64} \times \sqrt[3]{729} \\ \\ [/tex]
Now, Consider prime factorization of 64 and 729
[tex]\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:64 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:32\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:16\:\:}}\\\underline{\sf{2}}&\underline{\sf{\:\:8\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:4 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:2\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered} \\ [/tex]
[tex]\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} \\ \\ [/tex]
So, using these prime factors, we get
[tex]\sf \: = \: \sqrt[3]{\underbrace{2.2.2}.\underbrace{2.2.2}} \times \sqrt[3]{\underbrace{3.3.3}.\underbrace{3.3.3}} \\ \\ [/tex]
[tex]\sf \: = \: 2 \times 2 \times 3 \times 3 \\ \\ [/tex]
[tex]\sf \: = \: 36 \\ \\ [/tex]
Hence,
[tex]\sf \:\sf \: \implies \: \boxed{ \bf{ \: \sqrt[3]{64 \times 729} = 36 \: }} \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
[tex]\sf \: \sqrt[3]{ \frac{ - 512}{343} } \\ \\ [/tex]
[tex]\sf \: \: = - \: \sqrt[3]{ \frac{512}{343} } \\ \\ [/tex]
[tex]\sf \: \: = - \: \frac{ \sqrt[3]{512} }{ \sqrt[3]{343} } \\ \\ [/tex]
Now, Consider prime factorization of 512 and 343.
[tex]\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:512 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:256 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:128\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:64 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:32 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:16 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:8 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:4 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:2 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered} \\ [/tex]
[tex]\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} \\ [/tex]
[tex]\sf \: \: = - \: \frac{ \sqrt[3]{\underbrace{2.2.2}.\underbrace{2.2.2}.\underbrace{2.2.2}} }{ \sqrt[3]{\underbrace{7.7.7}} } \\ \\ [/tex]
[tex]\sf \: = \: - \: \frac{2 \times 2 \times 2}{7} \\ \\ [/tex]
[tex]\sf \: = \: - \: \frac{8}{7} \\ \\ [/tex]
[tex]\sf \: \sf \: \implies \: \boxed{ \bf{ \:\sqrt[3]{ \frac{ - 512}{343} } = - \frac{8}{7} \: }} \\ \\ \\ [/tex]
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Verified answer
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given expression is
[tex]\sf \: \sqrt[3]{64 \times 729} \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \sqrt[3]{64} \times \sqrt[3]{729} \\ \\ [/tex]
Now, Consider prime factorization of 64 and 729
[tex]\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:64 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:32\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:16\:\:}}\\\underline{\sf{2}}&\underline{\sf{\:\:8\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:4 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:2\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered} \\ [/tex]
[tex]\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} \\ \\ [/tex]
So, using these prime factors, we get
[tex]\sf \: = \: \sqrt[3]{\underbrace{2.2.2}.\underbrace{2.2.2}} \times \sqrt[3]{\underbrace{3.3.3}.\underbrace{3.3.3}} \\ \\ [/tex]
[tex]\sf \: = \: 2 \times 2 \times 3 \times 3 \\ \\ [/tex]
[tex]\sf \: = \: 36 \\ \\ [/tex]
Hence,
[tex]\sf \:\sf \: \implies \: \boxed{ \bf{ \: \sqrt[3]{64 \times 729} = 36 \: }} \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
Given expression is
[tex]\sf \: \sqrt[3]{ \frac{ - 512}{343} } \\ \\ [/tex]
[tex]\sf \: \: = - \: \sqrt[3]{ \frac{512}{343} } \\ \\ [/tex]
[tex]\sf \: \: = - \: \frac{ \sqrt[3]{512} }{ \sqrt[3]{343} } \\ \\ [/tex]
Now, Consider prime factorization of 512 and 343.
[tex]\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:512 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:256 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:128\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:64 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:32 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:16 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:8 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:4 \:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:2 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered} \\ [/tex]
[tex]\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} \\ [/tex]
So, using these prime factors, we get
[tex]\sf \: \: = - \: \frac{ \sqrt[3]{\underbrace{2.2.2}.\underbrace{2.2.2}.\underbrace{2.2.2}} }{ \sqrt[3]{\underbrace{7.7.7}} } \\ \\ [/tex]
[tex]\sf \: = \: - \: \frac{2 \times 2 \times 2}{7} \\ \\ [/tex]
[tex]\sf \: = \: - \: \frac{8}{7} \\ \\ [/tex]
Hence,
[tex]\sf \: \sf \: \implies \: \boxed{ \bf{ \:\sqrt[3]{ \frac{ - 512}{343} } = - \frac{8}{7} \: }} \\ \\ \\ [/tex]