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

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## 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) $$8100$$

Video Solution
Squares And Square Roots
Ex 6.3 | Question 4

## 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}

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