Solution

\[ \begin{align*} &= 3.5^3 \\ &= 3.5 \times 3.5 \times 3.5 \\ &= 12.25 \times 3.5 \\ &= 42.875 \end{align*} \]

Answer \( \color{red}3.5^3 = 42.875 \)

Solution

\[ \begin{array}{c|c} 2 & 392 \\ \hline 2 & 196 \\ \hline 2 & 98 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \end{array} \] \[ \begin{aligned} 392 &= 2^3 \times 7^2 \\ \text{Multiply by } & = 7 \\ \\ 7 \times 392 & = 2744 \\ \\ 2744 & = 2^3 \times 7^3 \\ & = (2 \times 7)^3 \\ 2744& = 14^3 \\ \end{aligned} \]

Answer \( \color{red}7 \)

Solution

\[ \begin{array}{c|c} 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \end{array} \] \[ \begin{aligned} 128 &= 2^3 \times 2^3 \times 2 \\ \text{Divide by } & = 2 \\ \\ 128 \div 2 & = 64 \\ \\ 64 & = 2^3 \times 2^3 \\ & = (2 \times 2)^3 \\ 64 & = 4^3 \\ \end{aligned} \]

Answer \( \color{red}2 \)

Solution

\[ \begin{array}{c|c} 2 & 13824 \\ \hline 2 & 6912 \\ \hline 2 & 3456 \\ \hline 2 & 1728 \\ \hline 2 & 864 \\ \hline 2 & 432 \\ \hline 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \end{array} \] \[ \begin{aligned} \sqrt[3]{13824} &= \sqrt[3]{2^3 \times 2^3 \times 2^3 \times 3^3} \\ &= 2 \times 2 \times 2 \times 3 \\ &= 4 \times 6 \\ \sqrt[3]{13824} &= 24 \end{aligned} \]

Answer \( \sqrt[3]{13824} = \color{red}24 \)

\[ \begin{aligned} \text{II} \quad & \quad \, \, \, \text{I} \\ 17 \quad & \quad 576 \\ \end{aligned} \] \[ \begin{aligned} \text{In the first group } 576 & \text{ has 6 in unit place.} \\ & \quad 6^3 = 216 \\ \therefore \text{ One's place } &= 6 \\[6pt] \end{aligned} \] \[ \begin{aligned} & \text{From the second group } 17 \\ & \quad 8 < 17 < 27 \\ & \quad 2^3 < 17 < 3^3 \\ \therefore \text{ Tens's place } &= 2 \\[6pt] & \sqrt[3]{17576} = 26 \end{aligned} \]

Answer \( \color{red} \sqrt[3]{17576} = 26 \)

Solution

\[ \begin{array}{c|c} 2 & 192 \\ \hline 2 & 96 \\ \hline 2 & 48 \\ \hline 2 & 24 \\ \hline 2 & 12 \\ \hline 2 & 6 \\ \hline 3 & 3 \\ \hline & 1 \end{array} \] \[ \begin{aligned} 192 &= 2^3 \times 2^3 \times 3\\ \text{Divide by } & = 3 \\ \\ 192 \div 3 &= 64 \\ \\ 64 &= 2^3 \times 2^3 \\ & = (2 \times 2)^3 \\ 64 & = 4^3 \\ \end{aligned} \]

Answer \( \color{red}3 \)

Solution

\[ \begin{array}{c|c} 2 & 474552 \\ \hline 2 & 237276 \\ \hline 2 & 118638 \\ \hline 3 & 59319 \\ \hline 3 & 19773 \\ \hline 3 & 6591 \\ \hline 13 & 2197 \\ \hline 13 & 169 \\ \hline 13 & 13 \\ \hline & 1 \end{array} \] \[ \begin{aligned} - \sqrt[3]{474552} &= - \left(\sqrt[3]{2^3 \times 3^3 \times 13^3}\right) \\ &= - \left( 2 \times 3 \times 13 \right) \\ &= - \left( 6 \times 13 \right) \\ - \sqrt[3]{474552} &= - 78 \end{aligned} \]

Answer \( - \sqrt[3]{474552} = \color{red}-78 \)

Solution

\[ \begin{aligned} \sqrt[3]{27} \times \sqrt[3]{2744} \end{aligned} \] \[ \begin{array}{cc} \begin{array}{c|c} 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} & \quad \begin{array}{c|c} 2 & 2744 \\ \hline 2 & 1372 \\ \hline 2 & 686 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array} \end{array} \] \[ \begin{aligned} & = \sqrt[3]{3^3} \times \sqrt[3]{2^3 \times 7^3} \\ & = 3 \times 2 \times 7 \\ & = 6 \times 7 \\ & = 42 \\ \end{aligned} \]

Answer \( \color{red}42\)

Solution

\[ \begin{aligned} &= \sqrt[3]{\frac{0.027}{0.008}} \div \sqrt{\frac{0.09}{0.04}} -1 \\ \\ & = \sqrt[3]{\frac{0.027 \times 1000}{0.008 \times 1000}} \div \sqrt{\frac{0.09 \times 100}{0.04 \times 100 }} -1 \\ \\ & = \sqrt[3]{\frac{27}{8}} \div \sqrt{\frac{9}{4}} -1 \\ \\ & = \sqrt[3]{\frac{3^3}{2^3}} \div \sqrt{\frac{3^2}{2^2}} -1 \\ \\ & = \frac{3}{2} \div \frac{3}{2} -1 \\ \\ & = \frac{\cancel3^1}{\cancel2_1} \times \frac{\cancel2^1}{\cancel3_1} -1 \\ \\ & = 1 - 1 \\ & = 0 \end{aligned} \]

Answer \( \color{red} 0\)

Solution

\[ \begin{aligned} &= \sqrt[3]{64 \times 729} \\[6pt] &= \sqrt[3]{4^3 \times 9^3} \\[6pt] &= 4 \times 9 \\[6pt] &= \color{green}36 \end{aligned} \]

Answer \( \color{red}36\)

Solution

\[ \begin{aligned} &= \sqrt[3]{1000} + \sqrt[3]{0.008} + \sqrt[3]{0.125} \\ \\ &= \sqrt[3]{10^3} + \sqrt[3]{\frac{8}{1000}} + \sqrt[3]{\frac{125}{1000}} \\ \\ &= 10 + \sqrt[3]{\frac{2^3}{10^3}} + \sqrt[3]{\frac{5^3}{10^3}} \\ \\ &= 10 + \frac{2}{10} + \frac{5}{10} \\ \\ &= 10 + 0.2 + 0.5 \\ &= 10 .7 \end{aligned} \]

Answer \( \color{red} 10.7\)

Solution

\[ \begin{aligned} \text{Let the numbers be } & 2x,3x,4x\\ (2x)^3+(3x)^3+(4x)^3 &= 33957\\ 8x^3+ 27x^3+ 64x^3 &= 33957\\ 99x^3 &= 33957 \\ \\ x^3 &= \frac{33957}{99} \\ \\ x^3 &= 343 \\ x &= \sqrt[3]{343} \\ x &= 7 \\ \\ \text{The numbers are} \\ 2x = 2 \times 7 & \implies 14 \\ 3x = 3 \times 7 & \implies 21 \\ 4x = 4 \times 7 & \implies 28 \\ \end{aligned} \]

Answer The numbers are \( \color{red} 14, 21, 28\)

Solution

\[ \begin{aligned} \text{Volume of a cube} & = 9261000 \ m^3 \\[5pt] \color{magenta}\text{Volume of a cube} &= \color{magenta} (Side)^3 \\[5pt] S^3 & = 9261000 \\[5pt] S & = \sqrt[3]{9261000} \\[5pt] S & = \sqrt[3]{9261 \times 1000} \\[5pt] \end{aligned} \] \[ \begin{array}{c|c} 3 & 9261 \\ \hline 3 & 3087 \\ \hline 3 & 1029 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \end{array} \] \[ \begin{aligned} S & = \sqrt[3]{9261 \times 1000} \\[5pt] S & = \sqrt[3]{3^3 \times 7^3 \times 10^3} \\[5pt] S & = 3 \times 7 \times 10 \\[5pt] S & = 210 \ m \\[5pt] \end{aligned} \]

Answer \( Side = \color{red} 210 \ m \)

Solution

\[ \begin{aligned} \text{Let the numbers be } & 2x,\,3x,\,4x\\ (2x)^3 + (3x)^3 + (4x)^3 &= 0.334125\\ 8x^3 + 27x^3 + 64x^3 &= 0.334125\\ 99x^3 &= 0.334125 \\ \\ x^3 &= \frac{0.334125}{99} \\ \\ x^3 &= \frac{0.334125 \times 1000000}{99 \times 1000000} \\ \\ x^3 &= \frac{\cancel{334125}^{3375}}{\cancel{99}_1 \times 1000000} \\ \\ x^3 &= \frac{3375}{1000000} \\ \\ x &= \sqrt[3]{\frac{3375}{1000000}} \\ \\ x &= \sqrt[3]{\frac{15^3}{100^3}} \\ \\ x &= \frac{15}{100} \\ \\ x &= 0.15 \\ \\ \text{The numbers are} \\ \\ 2x = 2 \times 0.15 & \implies 0.3 \\ 3x = 3 \times 0.15 & \implies 0.45 \\ 4x = 4 \times 0.15 & \implies 0.6 \\ \end{aligned} \]

Answer The numbers are \( \color{red}0.3, 0.45, 0.6\)

Solution

\[ \begin{array}{c|c} 2 & 8640 \\ \hline 2 & 4320 \\ \hline 2 & 2160 \\ \hline 2 & 1080 \\ \hline 2 & 540 \\ \hline 2 & 270 \\ \hline 3 & 135 \\ \hline 3 & 45 \\ \hline 3 & 15 \\ \hline 5 & 5 \\ \hline & 1 \end{array} \] \[ \begin{aligned} 8640 &= 2^3 \times 2^3 \times 3^3 \times 5 \\ \text{Divide by } & = 5 \\ \\ 8640 \div 5 & = 1728 \\ \\ 1728 & = 2^3 \times 2^3 \times 3^3 \\ & = (2 \times 2 \times 3)^3 \\ & = (4 \times 3)^3 \\ 1728 & = 12^3 \\ \end{aligned} \]

Answer Smallest number \( = \color{red} 5 \)

Solution

\[ \begin{aligned} \text{Let the number be } & = x \\ \text{Triple the number } & = 3x \\ \text{Cube of the number} & = (3x)^3 \\ \\ \text{To prove: } \color{magenta}(3x)^3 & = \color{magenta} 27 \times x^3 \\ \\ \text{Let }x & = 2 \\ LHS & = (3 \times 2)^3 \\ & = (6)^3 \\ & = 216 \\ \\ RHS & = 27 \times x^3 \\ & = 27 \times 2^3 \\ & = 27 \times 8 \\ & = 216 \\ \\ LHS & = RHS \end{aligned} \]

Solution

\[ \begin{aligned} \text{Surface area of a cube} &= 384\ \mathrm{m}^2 \\[5pt] \color{magenta}\text{Surface area of a cube} &= \color{magenta}6 \times (Side)^2 \\[5pt] 6 \times (Side)^2 &= 384 \\[5pt] (Side)^2 &= \frac{\cancel{384}^{64}}{\cancel6_1} \\[5pt] (Side)^2 &= 64 \\[5pt] Side &= \sqrt{64} \\[5pt] Side &= 8\ \mathrm{m} \\[10pt] \color{magenta}\text{Volume of the cube} &= \color{magenta}(Side)^3 \\[5pt] &= (8\ \mathrm{m})^3 \\[5pt] &= 512\ \mathrm{m}^3 \end{aligned} \]

Answer Volume of the cube \(= \color{red}512\ \mathrm{m}^3\)

Solution

\[ \begin{aligned} & = \left[\left(5^2 + 12^2\right)^{\frac{1}{2}}\right]^3 \\[6pt] & = \left[\left(25 + 144 \right)^{\frac{1}{2}}\right]^3 \\[6pt] & = \left[\left(169\right)^{\frac{1}{2}}\right]^3 \\[6pt] & = \left[\left(13^2 \right)^{\frac{1}{2}}\right]^3 \\[6pt] & = \left[\left(13\right)^{\cancel2^1 \times \frac{1}{\cancel2_1}}\right]^3 \\[6pt] & = 13^3 \\[6pt] & = 2197 \end{aligned} \]

Answer \(\color{red} 2197\)

Solution

\[ \begin{aligned} & = \left[\left(6^2 + 8^2\right)^{\frac{1}{2}}\right]^3 \\[6pt] & = \left[\left(36 + 64 \right)^{\frac{1}{2}}\right]^3 \\[6pt] & = \left[\left(100 \right)^{\frac{1}{2}}\right]^3 \\[6pt] & = \left[\left(10^2 \right)^{\frac{1}{2}}\right]^3 \\[6pt] & = \left[\left(10\right)^{\cancel2^1 \times \frac{1}{\cancel2_1}}\right]^3 \\[6pt] & = 10^3 \\[6pt] & = 1000 \end{aligned} \]

Answer \( \color{red} 1000 \)

Solution

\[ \begin{array}{c|c} 2 & 3600 \\ \hline 2 & 1800 \\ \hline 2 & 900 \\ \hline 2 & 450 \\ \hline 3 & 225 \\ \hline 3 & 75 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \end{array} \] \[ \begin{aligned} 3600 &= 2^3 \times 2 \times 3^2 \times 5^2 \\ \\ \text{Required no.} &= 2^2 \times 3 \times 5 \\ &= 4 \times 15 \\ \text{Multiply by } &= 60 \\ \\ 3600 \times 60 & = 216000 \\ \\ \sqrt[3]{216000} & = \sqrt[3]{2^3 \times 2^3 \times 3^3 \times 5^3} \\ & = 2 \times 2 \times 3 \times 5 \\ & = 4 \times 15 \\ \sqrt[3]{216000} & = 60 \\ \end{aligned} \]

Answer Smallest number \( = \color{red} 60\), Perfect cube \( = \color{red} 216000\) ,\(\sqrt[3]{216000} = \color{red} 60 \)

Solution

A perfect cube ending in 1 has cube root ending in 1.
Hence unit digit \(=1\).

Answer \( \color{red} 1\)

Solution

A perfect cube ending in 4 has cube root ending in 4 (since \(4^3=64\)).
Hence unit digit \(=4\).

Answer \( \color{red} 4\)

Solution

A perfect cube ending in 7 has cube root ending in 3 (since \(3^3=27\)).
Hence unit digit \(=3\).

Answer \( \color{red} 3\)

Solution

\[ \begin{aligned} & = \sqrt[3]{27000} \\ & = \sqrt[3]{27\times1000} \\ & = \sqrt[3]{3^3\times10^3} \\\ & = 3 \times 10 \\ & = 30 \end{aligned} \]

Answer \( \sqrt[3]{27000} = \color{red} 30\)