Question

$\huge{\mathfrak{\pink{☆Question ☆}}}$
Solve this
$\sqrt{ \frac{676}{6724} }$
​

Answer 1

$\sqrt{ \frac{676}{6724} } = 0.0003867091$

Answer 2

★ Concept :-

Here the concept of Prime Factorisation has been used. We see that we are give a term which we need to solve. We will solve this question in step - by - step method. Firstly by prime factorisation, we shall find the prime factors of the given numbers. Then we shall apply them and find the answer.

Let's do it !!

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★ To Solve :-

$\\\;\bf{\mapsto\;\;\red{\sqrt{\dfrac{676}{6724}}}}$

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★ Solution :-

Given,

$\\\;\sf{\mapsto\;\;\orange{\sqrt{\dfrac{676}{6724}}}}$

Let's start solving this.

• Step I ::

Here we shall firstly find the factors of the given term. So, by prime factorisation of the terms, we get

✒ 676 = 2 × 2 × 13 × 13 = 2² × 13²

✒ 6724 = 2 × 2 × 41 × 41 = 2² × 41²

• Step II ::

Here we square root the given terms using their respective prime factors.

$\\\;\sf{\rightarrow\;\;\sqrt{\dfrac{676}{6724}}\;=\;\bf{\sqrt{\dfrac{2^{2}\:\times\:13^{2}}{2^{2}\:\times\:41^{2}}}}}$

$\\\;\sf{\rightarrow\;\;\sqrt{\dfrac{676}{6724}}\;=\;\bf{\sqrt{\dfrac{2\:\times\:2\:\times\:13\:\times\:13}{2\:\times\:2\:\times\:41\:\times\:41}}}}$

Since we are given the squares of the numbers under square root. So we can take out the terms under the square root.

$\\\;\sf{\rightarrow\;\;\sqrt{\dfrac{676}{6724}}\;=\;\bf{\dfrac{2\:\times\:13}{2\:\times\:41}}}$

$\\\;\sf{\rightarrow\;\;\sqrt{\dfrac{676}{6724}}\;=\;\blue{\bf{\dfrac{26}{82}}}}$

• Step III ::

In this step, we shall simplify the given equation to more simplest term.

$\\\;\sf{\Longrightarrow\;\;\sqrt{\dfrac{676}{6724}}\;=\;\bf{\dfrac{26}{82}}}$

By prime factorisation of the given terms, we get

✒ 26 = 2 × 13

✒ 82 = 2 × 41

Here we see that 2 is a common term in prime factorisation of both the digits.

So cancelling the numerator and denominator by 2, we get

$\\\;\sf{\Longrightarrow\;\;\sqrt{\dfrac{676}{6724}}\;=\;\bf{\dfrac{26}{82}\:\times\:\dfrac{2}{2}}}$

$\\\;\sf{\Longrightarrow\;\;\sqrt{\dfrac{676}{6724}}\;=\;\bf{\dfrac{13}{41}}}$

This is the required answer.

$\\\;\underline{\boxed{\tt{Required\;\:answer\;=\;\bf{\purple{\dfrac{13}{41}}}}}}$

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