Ex.7.2 Q1 Cubes and Cube Roots - NCERT Maths Class 8

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Question

 Find the cube root of each of the following numbers by prime factorization method.

(i) \(64\)

(ii) \(512\)

(iii) \(10648\)

(iv) \(27000\)

(v) \(15625\)

(vi) \(13824\)

(vii) \(110592\)

(viii) \(46656\)

(ix) \(175616\)

(x) \(91125\)

Text Solution

Reasoning:

Factors in the prime factorization of cube should be grouped as triplets.

Steps:

(i)

\[\begin{align} 64&=\underline{2\times 2\times 2}\times \underline{2\times 2\times 2} \\ & ={{2}^{3}}\times {{2}^{3}} \\  \sqrt[3]{64}&=2\times 2=4  \end{align}\]

(ii) 

\[\begin{align}512&=\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{2\times 2\times 2} \\ & ={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}} \\  \sqrt[3]{512}&=2\times 2\times 2=8  \end{align}\]

(iii)

\[\begin{align}10648&=\underline{2\times 2\times 2}\times \underline{11\times 11\times 11} \\  & ={{2}^{3}}\times {{11}^{3}} \\  \sqrt[3]{10648}&=2\times 11=22  \end{align}\]

(iv) 

\[\begin{align} 27000&=\underline{2\times 2\times 2}\times \underline{3\times 3\times 3}\times \underline{5\times 5\times 5} \\ & ={{2}^{3}}\times {{3}^{3}}\times {{5}^{3}} \\ \sqrt[3]{27000}&=2\times 3\times 5=30  \end{align}\]

(v) 

\[\begin{align}15625&=\underline{5\times 5\times 5}\times \underline{5\times 5\times 5} \\  & ={{5}^{3}}\times {{5}^{3}} \\ \sqrt[3]{15625}&=5\times 5=25  \end{align}\]

(vi) 

\[\begin{align}13824&=\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{3\times 3\times 3} \\ & ={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{3}^{3}} \\  \sqrt[3]{13824}&=2\times 2\times 2\times 3=24  \end{align}\]

(vii)

\[\begin{align}110592&=\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{3\times 3\times 3} \\ & ={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{3}^{3}} \\ \sqrt[3]{110592}&=2\times 2\times 2\times 2\times 3=48  \end{align}\]

(viii)

\[\begin{align}46656& =\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{3\times 3\times 3}\times \underline{3\times 3\times 3} \\ & ={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{3}^{3}}\times {{3}^{3}} \\ \sqrt[3]{46656} & =2\times 2\times 3\times 3=36  \end{align}\]

(ix)

\[\begin{align}175616&=\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times \underline{7\times 7\times 7} \\ & ={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{7}^{3}} \\ \sqrt[3]{175616} &=2\times 2\times 2\times 7=56  \end{align}\]

(x)

\[\begin{align}91125&=\underline{5\times 5\times 5}\times \underline{3\times 3\times 3}\times \underline{3\times 3\times 3} \\ & ={{5}^{3}}\times {{3}^{3}}\times {{3}^{3}} \\  \sqrt[3]{91125}& =5\times 3\times 3=45  \end{align}\]

  
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