# 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) $$91125$$

Video Solution
Cubes And Cube Roots
Ex 7.2 | Question 1

## 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&=\begin{Bmatrix}\underline{2\times 2\times 2}\times \underline{3\times 3\times 3}\\\times \underline{5\times 5\times 5}\end{Bmatrix} \\ & ={{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&=\begin{Bmatrix}\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}\end{Bmatrix} \\ & ={{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&= \begin{Bmatrix} \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}\end{Bmatrix} \\ & ={{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& =\begin{Bmatrix}\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} \end{Bmatrix}\\ & ={{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&=\begin{Bmatrix}\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} \end{Bmatrix}\\ & ={{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&=\begin{Bmatrix}\underline{5\times 5\times 5}\times \underline{3\times 3\times 3}\\\times \underline{3\times 3\times 3}\end{Bmatrix} \\ & ={{5}^{3}}\times {{3}^{3}}\times {{3}^{3}} \\ \sqrt[3]{91125}& =5\times 3\times 3=45 \end{align}

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