Ex.6.3 Q4 Squares and Square Roots Solution - NCERT Maths Class 8

Go back to  'Ex.6.3'

Question

Find the square roots of the following numbers by the Prime Factorisation Method.

(i) \(729 \)

(ii) \(400 \)

(iii) \(1764\)

(iv) \(4096\)

(v) \(7744\)

(vi) \(9604 \)

(vii) \(5929 \)

(viii) \(9216\)

(ix) \(529\)

(x) \(8100\)

Text Solution

What is known?

Square

What is unknown?

Squareroot by using factorisation method.

Reasoning:

As we know that each prime factor in the prime factorisation of the square of a number, occurs twice the number of times it occurs in the prime factorisation of the number itself. Let us use this to find the square root of a given square number by pairing the prime factors.

Steps:

(i)

\(\begin{align}729 &= 3 \times 3 \times 3 \times 3 \times 3 \times 3\\\sqrt {729}  &= \sqrt {\underline {3 \times 3}  \times \underline {3 \times 3}  \times \underline {3 \times 3} } \\ &= 3 \times 3 \times 3\\ &= 27\end{align}\)

ii)

\(\begin{align}400 &= 2 \times 2 \times 2 \times 2 \times 5 \times 5\\ \sqrt {400}  &= \sqrt {\underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {5 \times 5} } \\&= 2 \times 2 \times 5\\&= 20\end{align}\)

ii)

\(\begin{align}1764 &= 2 \times 2 \times 3 \times 3 \times 7 \times 7\\\sqrt {1764}  &= \sqrt {\underline {2 \times 2}  \times \underline {3 \times 3}  \times \underline {7 \times7}} \\&= 2 \times 3 \times 7\\&= 42\end{align}\)

iv)

\(\begin{align}4096 &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\\\sqrt {4096}  &= \sqrt {\underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {2\times 2} } \\&= 2 \times 2 \times 2 \times 2 \times 2 \times 2\\&= 64\end{align}\)

\[\begin{align}&2\left| \!{\underline {\,{4096} \,}} \right. \\&2\left| \!{\underline {\,{2048} \,}} \right. \\&2\left| \!{\underline {\,{1024} \,}} \right. \\&2\left| \!{\underline {\,512 \,}} \right. \\&2\left| \!{\underline {\,256 \,}} \right.\\ &2\left| \!{\underline {\,128 \,}} \right.\\&2\left| \!{\underline {\,64\,}} \right.\\&2\left| \!{\underline {\,32 \,}} \right.\\&2\left| \!{\underline {\,16 \,}} \right.\\&2\left| \!{\underline {\,8\,}} \right.\\&2\left| \!{\underline {\,4 \,}} \right.\\ &\,\,\,\,\left| \!{\underline {\,2\,}} \right.\end{align}\]

(v)

\(\begin{align}7744 &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 \times 11\\  \sqrt {7744}  &= \sqrt {\underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {11 \times 11} } \\ &= 2 \times 2 \times 2 \times 11\\ &= 88\end{align}\)

\[\begin{align}&2\left| \!{\underline {\,{7744} \,}} \right. \\  &2\left| \!{\underline {\,{3872} \,}} \right. \\&2\left| \!{\underline {\,{1936} \,}} \right. \\&2\left| \!{\underline {\,968\,}} \right. \\&2\left| \!{\underline {\,484 \,}} \right.\\ &2\left| \!{\underline {\,242 \,}} \right.\\&2\left| \!{\underline {\,121\,}} \right.\\&\,\,\,\,\left| \!{\underline {\,11 \,}} \right.\end{align}\]

vi)

\(\begin{align}9604 &= 2 \times 2 \times 7 \times 7 \times 7 \times 7\\ \sqrt {9604}  &= \sqrt {\underline {2 \times 2}  \times \underline {7 \times 7}  \times \underline {7 \times 7} } \\&= 2 \times 7 \times 7\\&= 98\end{align}\)

[\begin{align}&2\left| \!{\underline {\,{9604} \,}} \right. \\
&2\left| \!{\underline {\,{4802} \,}} \right. \\&7\left| \!{\underline {\,{2401} \,}} \right. \\&7\left| \!{\underline {\,343\,}} \right. \\&7\left| \!{\underline {\,49\,}} \right.\\ &\,\,\,\,\left| \!{\underline {\,7\,}} \right.\end{align}\]

(vii)

\(\begin{align}5929 &= 7 \times 7 \times 11 \times 11\\\sqrt {5929}  &= \sqrt {\underline {7 \times 7}  \times \underline {11 \times 11} } \\ &= 7 \times 11\\&= 77\end{align}\)

iii)

\(\begin{align}9216 &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\\\sqrt {9216}  &= \sqrt {\underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {2 \times 2}  \times \underline {3 \times 3} } \\&= 2 \times 2 \times 2 \times 2 \times 2 \times 3\\&= 96\end{align}\)

\[\begin{align}&2\left| \!{\underline {\,{9216} \,}} \right. \\&2\left| \!{\underline {\,{4608} \,}} \right. \\&2\left| \!{\underline {\,{2304} \,}} \right. \\&2\left| \!{\underline {\,1152\,}} \right. \\&2\left| \!{\underline {\,576\,}} \right.\\ &2\left| \!{\underline {\,288\,}} \right.\\&2\left| \!{\underline {\,144\,}} \right.\\&2\left| \!{\underline {\,72\,}} \right.\\&2\left| \!{\underline {\,36\,}} \right.\\&2\left| \!{\underline {\,18\,}} \right.\\&3\left| \!{\underline {\,9\,}} \right.\\&\,\,\,\,\left| \!{\underline {\,3\,}} \right.\end{align}\]

ix)

\(\begin{align}529 &= 23 \times 23\\\sqrt {529}  &= \sqrt {\underline {23 \times 23} } \\ &= 23\end{align}\)

x)

\(\begin{align}8100 &= 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5\\ \sqrt {8100}  &= \sqrt {\underline {2 \times 2}  \times \underline {3 \times 3}  \times \underline {3 \times 3} \times \underline {5 \times 5} } \\& = 2 \times 3 \times 3 \times 5\\ &= 90\end{align}\)

\[\begin{align}&2\left| \!{\underline {\,{8100} \,}} \right. \\&2\left| \!{\underline {\,{4050} \,}} \right. \\&3\left| \!{\underline {\,{2025} \,}} \right. \\&3\left| \!{\underline {\,675\,}} \right. \\&3\left| \!{\underline {\,225\,}} \right.\\ &3\left| \!{\underline {\,75\,}} \right.\\&5\left| \!{\underline {\,25\,}} \right.\\&\,\,\,\,\left| \!{\underline {\,5\,}} \right.\end{align}\]