Question

square root
by prime factorization III) 7.29​

Answer 1

Answer:

729 is a perfect square number which is obtained by square of 27. Hence, the square root of 729 is a rational number.

Step-by-step explanation:

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Answer 2

$\large\underline{\sf{Solution-}}$

Given number is

$\purple{\rm :\longmapsto\: \sqrt{7.29}}$

can be rewritten as

$\purple{\rm \:  =  \: \sqrt{\dfrac{729}{100} } }$

Now, Let find the prime factors of 729 and 100

So, Consider

$\red{\rm :\longmapsto\:Prime \: factorization \: of \: 729}$

$\red{\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:729 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:243 \:\:}} \\\underline{\sf{3}}&\underline{\sf{\:\:81\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}}$

Thus,

$\red{\rm :\longmapsto\:Prime \: factorization \: of \: 729 =3.3.3.3.3.3}$

Now Consider

$\green{\rm :\longmapsto\:Prime \: factorization \: of \: 100}$

$\green{\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:100 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:50 \:\:}} \\\underline{\sf{5}}&\underline{\sf{\:\:25\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}} \\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}}$

Thus,

$\green{\rm :\longmapsto\:Prime \: factorization \: of \: 100 = 2.2.5.5}$

So, Now Consider

$\purple{\rm \:  =  \: \sqrt{\dfrac{729}{100} } }$

$\purple{\rm \:  =  \: \sqrt{\dfrac{ \underbrace{3.3} \: \underbrace{3.3} \: \underbrace{3.3}}{\underbrace{2.2} \: \underbrace{5.5}} } }$

$\purple{\rm \:  =  \: \dfrac{3 \times 3 \times 3}{2 \times 5} }$

$\purple{\rm \:  =  \: \dfrac{27}{10} }$

$\purple{\rm \:  =  \: 2.7}$

Hence,

$\purple{\rm \implies\:\boxed{ \tt{ \: \sqrt{7.29} \: = \: 2.7 \: }}}$

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More to know :-

1. The number of digits in square root of 2n digits is n.

2. The number of digits in square root of (2n + 1) digits is n.

3. Between squares of any two consecutive natural numbers n and n + 1 is always 2n.

4. Perfect square number can never end with the digit 2, 3, 7, 8 and odd number of zeroes.