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Answer :
(i) We have the prime factors of 64 as:
\(\qquad \begin{array}{c|lcr}
2 & 64\\
\hline
2 & 32\\
\hline
2 & 16\\
\hline
2 & 8\\
\hline
2 & 4\\
\hline
2 & 2\\
\hline
& 1\\
\end{array}
\)
\(256 =2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^3 \times 2^3\)
∴\(\sqrt[3]{64}=2 \times 2=4\)
Hence, the cube root of 64 is 4.
(ii) We have the prime factors of 512 as:
\(\qquad \begin{array}{c|lcr}
2 & 512\\
\hline
2 & 256\\
\hline
2 & 128\\
\hline
2 & 64\\
\hline
2 & 32\\
\hline
2 & 16\\
\hline
2 & 8\\
\hline
2 & 4\\
\hline
2 & 2\\
\hline
& 1\\
\end{array}
\)
\(256 =2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^3 \times 2^3 \times 2^3\)
∴\(\sqrt[3]{512}=2 \times 2 \times 2=8\)
Hence, the cube root of 512 is 8.
(iii) We have the prime factors of 10648 as:
\(\qquad \begin{array}{c|lcr}
2 & 10648\\
\hline
2 & 5324\\
\hline
2 & 2662\\
\hline
11 & 1331\\
\hline
11 & 121\\
\hline
11 & 11\\
\hline
& 1\\
\end{array}
\)
\(10648 =2 \times 2 \times 2 \times 11 \times 11 \times 11 = 2^3 \times 11^3 \times \)
∴\(\sqrt[3]{10648}=2 \times 11=22\)
Hence, the cube root of 10648 is 22.
(iv) We have the prime factors of 27000 as:
\(\qquad \begin{array}{c|lcr}
2 & 27000\\
\hline
2 & 13500\\
\hline
2 & 6750\\
\hline
3 & 3375\\
\hline
3 & 1125\\
\hline
3 & 375\\
\hline
5 & 125\\
\hline
5 & 25\\
\hline
5 & 5\\
\hline
& 1\\
\end{array}
\)
\(27000 =2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 = 2^3 \times 3^3 \times 5^3\)
∴\(\sqrt[3]{27000}=2 \times 3 \times 5=30\)
Hence, the cube root of 27000 is 30.
(v) We have the prime factors of 15625 as:
\(\qquad \begin{array}{c|lcr}
5 & 15625\\
\hline
5 & 3125\\
\hline
5 & 625\\
\hline
5 & 125\\
\hline
5 & 25\\
\hline
5 & 5\\
\hline
& 1\\
\end{array}
\)
\(15625 =5 \times 5 \times 5 \times 5 \times 5 \times 5= 5^3 \times 5^3\)
∴\(\sqrt[3]{15625}=5 \times 5=25\)
Hence, the cube root of 15625 is 25.
(vi) We have the prime factors of 13824 as:
\(\qquad \begin{array}{c|lcr}
2 & 13824\\
\hline
2 & 6912\\
\hline
2 & 3456\\
\hline
2 & 1728\\
\hline
2 & 864\\
\hline
2 & 432\\
\hline
2 & 216\\
\hline
2 & 108\\
\hline
2 & 54\\
\hline
3 & 27\\
\hline
3 & 9\\
\hline
3 &3 \\
\hline
& 1\\
\end{array}
\)
\(13824=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \)
\(\Rightarrow 2^3 \times 2^3 \times 2^3 \times 3^3\)
∴\(\sqrt[3]{13824}=2 \times 2 \times 2 \times 3=24\)
Hence, the cube root of 13824 is 24.
(vii) We have the prime factors of 110592 as:
\(\qquad \begin{array}{c|lcr}
2 & 110592\\
\hline
2 & 55296\\
\hline
2 & 27648\\
\hline
2 & 13824\\
\hline
2 & 6912\\
\hline
2 & 3456\\
\hline
2 & 1728\\
\hline
2 & 864\\
\hline
2 & 432\\
\hline
2 & 216\\
\hline
2 & 108\\
\hline
2 & 54\\
\hline
3 & 27\\
\hline
3 & 9\\
\hline
3 &3 \\
\hline
& 1\\
\end{array}
\)
\(110592=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \)
\(\Rightarrow 2^3 \times 2^3 \times 2^3 \times 2^3 \times 3^3\)
∴\(\sqrt[3]{110592}=2 \times 2 \times 2 \times 2 \times 3=48\)
Hence, the cube root of 110592 is 48.
(viii) We have the prime factors of 46656 as:
\(\qquad \begin{array}{c|lcr}
2 & 46656\\
\hline
2 & 23328\\
\hline
2 & 11664\\
\hline
2 & 5832\\
\hline
2 & 2916\\
\hline
2 & 1458\\
\hline
3 & 729\\
\hline
3 & 243\\
\hline
3 & 81\\
\hline
3 & 27\\
\hline
3 & 9\\
\hline
3 &3 \\
\hline
& 1\\
\end{array}
\)
\(46656=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)
\(\Rightarrow 2^3 \times 2^3 \times 3^3 \times 3^3\)
∴\(\sqrt[3]{46656}=2 \times 2 \times 3 \times 3=36\)
Hence, the cube root of 46656 is 36.
(ix) We have the prime factors of 175616 as:
\(\qquad \begin{array}{c|lcr}
2 & 175616\\
\hline
2 & 87808\\
\hline
2 & 43904\\
\hline
2 & 21952\\
\hline
2 & 10976\\
\hline
2 & 5488\\
\hline
2 & 2744\\
\hline
2 & 1372\\
\hline
2 & 686\\
\hline
7 & 343\\
\hline
7 & 49\\
\hline
7 &7 \\
\hline
& 1\\
\end{array}
\)
\(175616=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 7 \)
\(\Rightarrow 2^3 \times 2^3 \times 2^3 \times 7^3\)
∴\(\sqrt[3]{175616}=2 \times 2 \times 2 \times 7 =56\)
Hence, the cube root of 175616 is 56.
(x) We have the prime factors of 91125 as:
\(\qquad \begin{array}{c|lcr}
3 & 91125\\
\hline
3 & 30375\\
\hline
3 & 10125\\
\hline
3 & 3375\\
\hline
3 & 1125\\
\hline
3 & 375\\
\hline
5 & 125\\
\hline
5 & 25\\
\hline
5 & 5\\
\hline
& 1\\
\end{array}
\)
\(91125 =3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 = 3^3 \times 3^3 \times 5^3\)
∴\(\sqrt[3]{91125}=3 \times 3 \times 5=45\)
Hence, the cube root of 91125 is 45.