# 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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