Ex.6.4 Q2 Squares and Square Roots - NCERT Maths Class 8

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Question

Find the number of digits in the square root of each of the following numbers (without any calculation).

(i) \(64\)             

(ii) \(144\)             

(iii) \(4489\)         

(iv) \(27225\)         

(v) \(390625\)

Text Solution

What is known?

Squares

What is unknown?

Number of digits in square root

Reasoning:

If a perfect square is of n digits then it’s square root will have \(\begin{align}\frac{n}{2}\end{align}\) digits, if n is even and \(\begin{align}\frac{{\left( {n + 1} \right)}}{2}\end{align}\) if \(n\) is odd

Steps

(i) \(64\)

\(n = 2 \rm(even) \)

\(\begin{align}{\rm{Number}}\;{\rm{of}}\;{\rm{digits}} = \frac{2}{2} = 1\end{align}\)

 (ii) \(144\)

\(n = 3\)

\(\begin{align}{\rm{Number}}\;{\rm{of}}\;{\rm{digits}} &= \frac{{n + 1}}{2}\\ &= \frac{{3 + 1}}{2} \\&= \frac{4}{2} = 2\end{align}\)

 (iii) \(4489\)

\(n = 4\)

\(\begin{align}{\rm{Number}}\;{\rm{of}}\;{\rm{digits}} = \frac{n}{2} = \frac{4}{2} = 2\end{align}\)

(iv) \(27225 \)

\(n = 5\)

\(\begin{align}{\rm{Number}}\;{\rm{of}}\;{\rm{digits}} &= \frac{{n + 1}}{2}\\ &= \frac{{5 + 1}}{2} \\&= \frac{6}{2} = 3\end{align}\)

(v) \(390625 \)

\(n = 6\)

\(\begin{align}{\rm{Number}}\;{\rm{of}}\;{\rm{digits}} = \frac{n}{2} = \frac{6}{2} = 3\end{align}\)