Ex.7.1 Q1 Cubes and Cube Roots - NCERT Maths Class 8

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Question

Which of the following numbers are not perfect cubes?

(i) \(216\)         (ii) \(128\)         (iii) \(1000\)         (iv) \(100\)         (v) \(46656\)

Text Solution

 

What is unknown?

To find the numbers which are not perfect cubes.

Reasoning:

A number is a prefect cube only when each factor in the prime factorization is grouped in triples.

Steps: 

(i)

\[\begin{align}216 &= {2 \times 2 \times 2} \times {3 \times 3 \times 3} \\&= {2^3} \times {3^3}\\& ={{(2\times 3)}^{3}} \\& ={{6}^{3}} \\\end{align}\]

\(\therefore 216\) is a perfect cube

 (ii)

\[\begin{align}128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \end{align}\]

\(128 = {2^3} \times {2^3} \times 2\) one of the \(2\) is not grouped in triples

\(\therefore \;128\) is not a perfect cube

 (iii)   

\[\begin{align}1000 &= 2 \times 2 \times 2 \times 5 \times 5 \times 5\\
1000 &= {2^3} \times {5^3}\end{align}\]

\(\therefore\;1000\) is a perfect cube

(iv) 

\[\begin{align}100&= 2 \times 2 \times 5 \times 5\\&= {2^2} \times {5^2}\end{align}\]

Both \(2\) and \(5\) are not grouped in triples

\(\therefore \;100\) is not a perfect cube.

(iv) 

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

\(\therefore\;46656\) is a perfect cube

  
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