Question

Find the square root of $33.64$.

Answer 1

Hint: The given number is a decimal number and to find the square root firstly convert the decimal number in to fraction and find the square roots of numerator and denominator separately either by prime factorization or by division method then divide their square roots to find the square root of the given decimal number.

Complete step-by-step answer:
Here, the given decimal number is $33.64$.
By converting this decimal number in to fraction we get $\dfrac{{3364}}{{100}}$
Now, we have to find the square root of $3364$ by the prime factorization method.
Prime factors of number $3364$ is
$
\Rightarrow 3364 = 2 \times 2 \times 29 \times 29 \\
\Rightarrow 3364 = {2^2} \times {29^2} \\
 $
Divide the power of each numeral in the prime factor of the given number by $2$to get the square root.
$
\Rightarrow \sqrt {3364} = {2^{\dfrac{2}{2}}} \times {29^{\dfrac{2}{2}}} \\
\Rightarrow \sqrt {3364} = 2 \times 29 \\
\Rightarrow \sqrt {3364} = 58
 $
Now, similarly find the square root of $100$ by prime factorization method.
Prime factor of number $100$ is
$100 = 2 \times 2 \times 5 \times 5$
$100 = {2^2} \times {5^2}$
Similarly, divide the power of each numeral in the prime factor of the given number by $2$to get the square root.
$
\Rightarrow \sqrt {100} = {2^{\dfrac{2}{2}}} \times {5^{\dfrac{2}{2}}} \\
\Rightarrow \sqrt {100} = 2 \times 5 \\
\Rightarrow \sqrt {100} = 10
 $
Now, by dividing the square root of $3364$ by the square root of $100$ we get,
$\sqrt {33.64} = \dfrac{{\sqrt {3364} }}{{\sqrt {100} }} = \dfrac{{58}}{{10}} = 5.8$

Thus, the square root of the given decimal number is $5.8$

Note: This method of prime factorization is also used for finding the cube root of the given decimal numbers. But instead of dividing the power of each numerals in prime factor of the given number by $2$ divide it by $3$ to get perfect cube root of the numerator and the denominator of the converted fraction then divide their cube roots to get the cube root of the given decimal numbers.