Question

Find the cube root of 110592.


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Answer 1

Cube roots :-

If n is a perfect cube, then for some integer m, n = m³. This implies that m is the cube root of n.

For example, (i) 8 is a cube root of 512

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎because 8³ = 512 [8 × 8 × 8 = 8³]

(ii) 3 is a cube root of 27

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎because 3³ = 27 [3 × 3 × 3 = 3³]

If m is a cube root of n, then we can write m = $\sqrt[3]{n}$

i.e., = $\sqrt[3]{512} = 8$ and $\sqrt[3]{27} = 3$

We use the symbol $\sqrt[3]{ \: \: }$ to denote cube root.

Cube Root By Prime Factorisation

The prime factorisation of a perfect cube prime occurs in triples.

∴ We can find $\sqrt[3]{n}$ by using the following algorithm:

Step 1 : Find the prime factorisation of n.

Step 2 : Group the factors in triples such that all three factors in each triple are the same.

Step 3 : If some prime factors are left ungrouped, the number n is not a perfect cube and then the process stops.

Step 4 : If no factor is left ungrouped, choose one factor from each group and take their product. The product is the cube root of n.

So let's answer the question.

To find the Cube root of 110592.

Note : Use the given algorithm ↑ to find the cube root of 110592.

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {{ \underline{\sf{ \: \: 2 \: \: }}}}&{{ \underline{\sf{\:\:110592\:\:}}}}\\ {{ \underline{\sf{ \: \: 2 \: \: }}}}& { \underline{\sf{\:\:55296 \:\:}}} \\{ \underline{\sf{ \: \: 2 \: \: }}}&{ \underline{\sf{\:\:27648\:\:}}}\\ {{ \underline{\sf{ \: \: 2 \: \: }}}}& { \underline{\sf{\:\:13824 \:\:}}} \\ {{ \underline{\sf{ \: \: 2\: \: }}}}& { \underline{\sf{\:\:6912\:\:}}}\\ {{ \underline{\sf{ \: \: 2 \: \: }}}}& { \underline{\sf{\:\:3456\:\:}}}\\ { \underline{\sf{ \: \: 2\: \: }}}& { \underline{\sf{\:\:1728 \:\:}}} \\ { \underline{\sf{ \: \: 2\: \: }}}& { \underline{\sf{\:\:864 \:\:}}} \\ { \underline{\sf{ \: \: 2\: \: }}}& { \underline{\sf{\:\:432\:\:}}}\\ { \underline{\sf{ \: \: 2\: \: }}}& { \underline{\sf{\:\:216\:\:}}}\\ { \underline{\sf{ \: \: 2\: \: }}}& { \underline{\sf{\:\:108 \:\:}}}\\ { \underline{\sf{ \: \: 2\: \: }}}& { \underline{\sf{\:\:54 \:\:}}}\\ { \underline{\sf{ \: \: 3\: \: }}}& { \underline{\sf{\:\:27\:\:}}}\\ { \underline{\sf{ \: \: 3\: \: }}}& { \underline{\sf{\:\:9 \:\:}}}\\ { \underline{\sf{ \: \: 3\: \: }}}& { \underline{\sf{\:\:3\:\:}}}\\ { \sf{ \: \: \: \: }}& { \sf{\:\:1 \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

$\sf \therefore 110592 = \underbrace{ 2 \times 2 \times 2}_{2} \times\underbrace{ 2 \times 2 \times 2}_{2} \times\underbrace{ 2 \times 2 \times 2}_{2} \times \underbrace{ 2 \times 2 \times 2}_{2} \times \underbrace{3 \times 3 \times 3}_{3}$

$\sf\sqrt[3]{110592} = 2 \times 2 \times 2 \times 2 \times 3 = 48$

∴ The Cube Root of 110592 = 48