\[ \begin{aligned} & \text{Prime factorisation of } 5832 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 5832 \\ \hline 2 & 2916 \\ \hline 2 & 1458 \\ \hline 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} \sqrt[3]{5832} &= \sqrt[3]{2^3 \times 3^3 \times 3^3} \\ \\ &= \sqrt[3]{(2 \times 3 \times 3)^3} \\ \\ &= 2 \times 3 \times 3 \\ \\ \color{green} \sqrt[3]{5832} &= \color{green} 18 \\ \\ \end{aligned} \]

Answer \( \color{red} \sqrt[3]{5832} = 18 \)

\[ \begin{aligned} & \text{Prime factorisation of } 1728 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 1728 \\ \hline 2 & 864 \\ \hline 2 & 432 \\ \hline 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} \sqrt[3]{1728} &= \sqrt[3]{2^3 \times 3^3 \times 3^3} \\ \\ &= \sqrt[3]{(2 \times 2 \times 3)^3} \\ \\ &= 2 \times 2 \times 3 \\ \\ \color{green} \sqrt[3]{1728} &= \color{green} 12 \\ \\ \end{aligned} \]

Answer \( \color{red} \sqrt[3]{1728} = 12 \)

\[ \begin{aligned} & \text{Prime factorisation of } 216000 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 216000 \\ \hline 2 & 108000 \\ \hline 2 & 54000 \\ \hline 2 & 27000 \\ \hline 2 & 13500 \\ \hline 2 & 6750 \\ \hline 3 & 3375 \\ \hline 3 & 1125 \\ \hline 3 & 375 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} \sqrt[3]{216000} &= \sqrt[3]{2^3 \times 2^3 \times 3^3 \times 5^3} \\ \\ &= \sqrt[3]{(2 \times 2 \times 3 \times 5)^3} \\ \\ &= 2 \times 2 \times 3 \times 5 \\ \\ \color{green} \sqrt[3]{216000} &= \color{green} 60 \\ \\ \end{aligned} \]

Answer \( \color{red} \sqrt[3]{216000} = 60 \)

\[ \begin{aligned} & \text{Prime factorisation of } 21952 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 21952 \\ \hline 2 & 10976 \\ \hline 2 & 5488 \\ \hline 2 & 2744 \\ \hline 2 & 1372 \\ \hline 2 & 686 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} \sqrt[3]{21952} &= \sqrt[3]{2^3 \times 2^3 \times 7^3} \\ \\ &= \sqrt[3]{(2 \times 2 \times 7)^3} \\ \\ &= 2 \times 2 \times 7 \\ \\ \color{green} \sqrt[3]{21952} &= \color{green} 28 \\ \\ \end{aligned} \]

Answer \( \color{red} \sqrt[3]{21952} = 28 \)

\[ \begin{aligned} &\sqrt[3]{-1728} = -\sqrt[3]{1728} \\ \\ & \text{Prime factorisation of } 1728 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 1728 \\ \hline 2 & 864 \\ \hline 2 & 432 \\ \hline 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} -\sqrt[3]{1728} &= -\sqrt[3]{2^3 \times 2^3 \times 3^3} \\ &= -\sqrt[3]{(2 \times 2 \times 3)^3} \\ &= -(2 \times 2 \times 3) \\ &= - (4 \times 3) \\ &= \color{green} -12 \end{aligned} \]

Answer \( \color{red} - \sqrt[3]{1728} = -12 \)

\[ \begin{aligned} &\sqrt[3]{-2744000} = -\sqrt[3]{2744000} \\ \\ & \text{Prime factorisation of } 2744000 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 2744000 \\ \hline 2 & 1372000 \\ \hline 2 & 686000 \\ \hline 2 & 343000 \\ \hline 2 & 171500 \\ \hline 2 & 85750 \\ \hline 5 & 42875 \\ \hline 5 & 8575 \\ \hline 5 & 1715 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} -\sqrt[3]{2744000} &= -\sqrt[3]{2^3 \times 2^3 \times 5^3 \times 7^3} \\ &= -\sqrt[3]{(2 \times 2 \times 5 \times 7)^3} \\ &= -(2 \times 2 \times 5 \times 7) \\ &= -(4 \times 5 \times 7) \\ &= -(20 \times 7) \\ &= \color{green} -140 \end{aligned} \]

Answer \( \color{red} - \sqrt[3]{2744000} = -140 \)

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

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

\[ \begin{aligned} &\sqrt[3]{-5832} = -\sqrt[3]{5832} \\ \\ & \text{Prime factorisation of } 5832 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 5832 \\ \hline 2 & 2916 \\ \hline 2 & 1458 \\ \hline 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} -\sqrt[3]{5832} &= -\sqrt[3]{2^3 \times 3^3 \times 3^3} \\ &= -\sqrt[3]{(2 \times 3 \times 3)^3} \\ &= -(2 \times 3 \times 3) \\ &= -(2 \times 9) \\ &= \color{green} -18 \end{aligned} \]

Answer \( \color{red} - \sqrt[3]{5832} = -18 \)

\[ \begin{aligned} &= \sqrt[3]{8 \times 125} \\ \\ &= \sqrt[3]{8} \times \sqrt[3]{ 125} \\ \\ \end{aligned} \] \[ \begin{array}{cc} \begin{array}{c|c} 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} & \quad \begin{array}{c|c} 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \end{array} \] \[ \begin{aligned} &= \sqrt[3]{2^3} \times \sqrt[3]{5^3} \\ &= 2 \times 5 \\ &= \color{green} 10 \end{aligned} \]

Answer \( \color{red} \sqrt[3]{8 \times 125} = 10 \)

\[ \begin{aligned} &= \sqrt[3]{3375 \times (-729)} \\ \\ &= \sqrt[3]{3375} \times \left(-\sqrt[3]{729}\right) \\ \\ \end{aligned} \] \[ \begin{array}{cc} \begin{array}{c|c} 3 & 3375 \\ \hline 3 & 1125 \\ \hline 3 & 375 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} & \quad \begin{array}{c|c} 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \end{array} \] \[ \begin{aligned} &= \sqrt[3]{3^3 \times 5^3} \times \left( -\sqrt[3]{3^3 \times 3^3} \right) \\ &= 3 \times 5 \times (-3 \times 3) \\ &= 15 \times (-9) \\ &= \color{green} -135 \end{aligned} \]

Answer \( \color{red} \sqrt[3]{3375 \times (-729)} = -135 \)

\[ \begin{aligned} &= \sqrt[3]{4^3 \times 5^3} \\ &= \sqrt[3]{4^3} \times \sqrt[3]{5^3} \\ &= 4 \times 5 \\ &= \color{green} 20 \end{aligned} \]

Answer \( \color{red} \sqrt[3]{4^3 \times 5^3} = 20 \)

\[ \begin{aligned} &= \frac{\sqrt[3]{4913}}{\sqrt[3]{3375}} \\ \\ \end{aligned} \] \[ \begin{array}{cc} \begin{array}{c|c} 17 & 4913 \\ \hline 17 & 289 \\ \hline 17 & 17 \\ \hline & 1 \\ \end{array} & \quad \begin{array}{c|c} 3 & 3375 \\ \hline 3 & 1125 \\ \hline 3 & 375 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \end{array} \] \[ \begin{aligned} &= \frac{\sqrt[3]{17 ^3}}{\sqrt[3]{3^3 \times 5^3}} \\ \\ &= \frac{17}{3 \times 5} \\ \\ &= \frac{17}{15} \\ \\ &= \color{green} 1\frac{2}{15} \end{aligned} \]

Answer \( \displaystyle \color{red} \sqrt[3]{\frac{4913}{3375}} = 1\frac{2}{15} \)

\[ \begin{aligned} &= \frac{-\sqrt[3]{512}}{\sqrt[3]{343}} \\ \\ \end{aligned} \] \[ \begin{array}{cc} \begin{array}{c|c} 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} & \quad \begin{array}{c|c} 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array} \end{array} \] \[ \begin{aligned} &= \frac{-\sqrt[3]{2^3 \times 2^3 \times 2^3}}{\sqrt[3]{7^3}} \\ \\ &= \frac{-(2 \times 2 \times 2)}{7} \\ \\ &= \frac{-8}{7} \\ \\ &= \color{green} -1\frac{1}{7} \\ \\ \end{aligned} \]

Answer \( \displaystyle \color{red} \sqrt[3]{\frac{512}{343}} = -1\frac{1}{7} \)

\[ \begin{aligned} &= \frac{\sqrt[3]{\cancel{686}^{343}}}{\sqrt[3]{\cancel{2662}_{1331}}} \\ \\ &= \frac{\sqrt[3]{343}}{\sqrt[3]{1331}} \\ \\ \end{aligned} \] \[ \begin{array}{cc} \begin{array}{c|c} 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array} & \quad \begin{array}{c|c} 11 & 1331 \\ \hline 11 & 121 \\ \hline 11 & 11 \\ \hline & 1 \\ \end{array} \end{array} \] \[ \begin{aligned} &= \frac{\sqrt[3]{7^3}}{\sqrt[3]{11^3}} \\ \\ &= \color{green} \frac{7}{11} \end{aligned} \]

Answer \( \displaystyle \color{red} \sqrt[3]{\frac{686}{2662}} = \frac{7}{11} \)

\[ \begin{aligned} & \text{Prime factorisation of } 5400 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 5400 \\ \hline 2 & 2700 \\ \hline 2 & 1350 \\ \hline 3 & 675 \\ \hline 3 & 225 \\ \hline 3 & 75 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} & 5400 = 2^3 \times 3^3 \times 5 \times 5 \\ & \text{We need to multiply 5400 by } 5 \text{ to make it a perfect cube.} \\ \end{aligned} \]

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

\[ \begin{aligned} & \text{Prime factorisation of } 16384 \\ \end{aligned} \] \[ \begin{array}{c|c} 2 & 16384 \\ \hline 2 & 8192 \\ \hline 2 & 4096 \\ \hline 2 & 2048 \\ \hline 2 & 1024 \\ \hline 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} & 16384 = 2^3 \times 2^3 \times 2^3 \times 2^3 \times 2 \times 2 \\ & \text{So, we need to divide 16384 by } 2^2 = 4 \\ \end{aligned} \]

Answer Required number \( = \color{red} 4 \)

\[ \begin{aligned} \text{II} \quad & \quad \, \, \, \text{I} \\ 10 \quad & \quad 648 \\ \end{aligned} \] \[ \begin{aligned} \text{In the first group } 648 & \text{ has 8 in unit place.} \\ & 2^3 = 8 \\ \therefore \text{ One's place } & = 2 \\ \\ \end{aligned} \] \[ \begin{aligned} & \text{From the second group } 10 \\ & \text{We know that } \\ & 8 < 10 < 27 \\ & 2^3 < 10 < 3^3 \\ \therefore \, & \, \text{ Tens's place } = 2 \\ \\ &\sqrt[3]{10648} = 22 \end{aligned} \]

Answer \( \color{red} \sqrt[3]{10648} = 22 \)

\[ \begin{aligned} \text{II} \quad & \quad \, \, \, \text{I} \\ 15 \quad & \quad 625 \\ \end{aligned} \] \[ \begin{aligned} \text{In the first group } 625 & \text{ has 5 in unit place.} \\ & 5^3 = 125 \\ \therefore \text{ One's place } & = 5 \\ \\ \end{aligned} \] \[ \begin{aligned} & \text{From the second group } 15 \\ & \text{We know that } \\ & 8 < 15 < 27 \\ & 2^3 < 15 < 3^3 \\ \therefore \, & \, \text{ Tens's place } = 2 \\ \\ &\sqrt[3]{15625} = 25 \end{aligned} \]

Answer \( \color{red} \sqrt[3]{15625} = 25 \)

\[ \begin{aligned} \text{II} \quad & \quad \, \, \, \text{I} \\ 110 \quad & \quad 592 \\ \end{aligned} \] \[ \begin{aligned} \text{In the first group } 592 & \text{ has 2 in unit place.} \\ & 8^3 = 512 \\ \therefore \text{ One's place } & = 8 \\ \\ \end{aligned} \] \[ \begin{aligned} & \text{From the second group } 110 \\ & \text{We know that } \\ & 64 < 110 < 125 \\ & 4^3 < 110 < 5^3 \\ \therefore \, & \, \text{ Tens's place } = 4 \\ \\ &\sqrt[3]{110592} = 48 \end{aligned} \]

Answer \( \color{red} \sqrt[3]{110592} = 48 \)

\[ \begin{aligned} \text{II} \quad & \quad \, \, \, \text{I} \\ 91 \quad & \quad 125 \\ \end{aligned} \] \[ \begin{aligned} \text{In the first group } 125 & \text{ has 5 in unit place.} \\ & 5^3 = 125 \\ \therefore \text{ One's place } & = 5 \\ \\ \end{aligned} \] \[ \begin{aligned} & \text{From the second group } 91 \\ & \text{We know that } \\ & 64 < 91 < 125 \\ & 4^3 < 91 < 5^3 \\ \therefore \, & \, \text{ Tens's place } = 4 \\ \\ &\sqrt[3]{91125} = 45 \end{aligned} \]

Answer \( \color{red} \sqrt[3]{91125} = 45 \)