3.Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

(i) 81

(ii) 128

(iii) 135

(iv) 92

(v) 704

(i) 81

(ii) 128

(iii) 135

(iv) 92

(v) 704

(i) We have the prime factors of 81 as:

\(\qquad \begin{array}{c|lcr}
3 & 81\\
\hline
3 & 27\\
\hline
3 & 9\\
\hline
3 & 3\\
\hline
& 1
\end{array}
\)

\(81 = 3 \times 3 \times 3 \times 3 = 3^3 \times 3\)

We can observe clearly that, number 3 is divided to 81 to make it a perfect cube.

So, 81 ÷ 3 = 27 which is a perfect cube.
Thus, the required smallest number to be divided is 3.

(ii) We have the prime factors of 256 as:

\(\qquad \begin{array}{c|lcr}
2 & 128\\
\hline
2 & 64\\
\hline
2 & 32\\
\hline
2 & 16\\
\hline
2 & 8\\
\hline
2 & 4\\
\hline
2 & 2\\
\hline
& 1\\
\end{array}
\)

\(128 =2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^3 \times 2^3 \times 2\)

We can observe clearly that, number 2 is divided to 128 to make it a perfect cube.

So, 128 ÷ 2 = 64 which is a perfect cube.
Thus, the required smallest number to be divided is 2.

(iii) We have the prime factors of 135 as:

\(\qquad \begin{array}{c|lcr}
3 & 135\\
\hline
3 & 45\\
\hline
3 & 15\\
\hline
5 & 5\\
\hline
& 1
\end{array}
\)

\(81 = 3 \times 3 \times 3 \times 5 = 3^3 \times 5\)

We can observe clearly that, number 5 is divided to 135 to make it a perfect cube.

So, 135 ÷ 5= 27 which is a perfect cube.
Thus, the required smallest number to be divided is 5.

(iv) We have the prime factors of 192 as:

\(\qquad \begin{array}{c|lcr}
2 & 192\\
\hline
2 & 96\\
\hline
2 & 48\\
\hline
2 & 24\\
\hline
2 & 12\\
\hline
2 & 6\\
\hline
3 & 3\\
\hline
& 1\\
\end{array}
\)

\(192 =2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^3 \times 2^3 \times 3\)

We can observe clearly that, number 3 is divided to 192 to make it a perfect cube.

So, 192 ÷ 3 = 64 which is a perfect cube.
Thus, the required smallest number to be divided is 3.

(v) We have the prime factors of 704 as:

\(\qquad \begin{array}{c|lcr}
2 & 704\\
\hline
2 & 352\\
\hline
2 & 176\\
\hline
2 & 88\\
\hline
2 & 44\\
\hline
2 & 22\\
\hline
11 & 11\\
\hline
& 1\\
\end{array}
\)

\(128 =2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 = 2^3 \times 2^3 \times 11\)

We can observe clearly that, number 11 is divided to 704 to make it a perfect cube.

So, 704 ÷ 11 = 64 which is a perfect cube.
Thus, the required smallest number to be divided is 11.