Question

check wheather the number 600 and 729 are perfect square or note?​

Answer 1

Answer :-

• 600 is not a perfect square.
• 729 is a perfect square.

To find :-

• Whether the numbers 600 and 729 are perfect squares.

Solution :-

• First let's check whether 600 is a perfect square!

Prime factorisation of 600 :-

$\begin{array}{c | c} \underline2 & \underline{600} \\ \underline2&\underline{300} \\ \underline2&\underline{150} \\ \underline3 &\underline{75} \\ \underline5&\underline{25} \\ \underline5&\underline5 \\ &1\end{array}$

The given number is 600.

It can be expressed as :-

$\sf \underbrace{2 \times 2} \times 2 \times 3 \times \underbrace{5 \times 5}$

We see that :-

• 2 and 3 are left unpaired.

Hence :-

• 600 is not a perfect square.

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• Now let's check whether 729 is a perfect square!

Prime factorisation of 729 :-

$\begin{array}{c | c} \underline3 &\underline{729} \\ \underline3&\underline{243} \\ \underline3 &\underline{81}\\ \underline3&\underline{27} \\ \underline3&\underline9 \\ \underline3&\underline3 \\ &1\end{array}$

The given number is 729.

It can be expressed as :-

$\sf \underbrace{3 \times 3} \times \underbrace{3 \times 3} \times \underbrace{3 \times 3}$

Since :-

• 729 can be expressed as the product of pairs of equal prime factors.

Hence :-

• 729 is a perfect square.

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