Question

What is the square root of \[6.25\]?

Answer 1

Hint: A square root of a number is a number which on multiplying itself yields the original number. In other words, let \[x\] be a number and \[y\] be a square root of \[x\] then \[x = y \times y = {y^2}\]. Here we are asked to find the square root of a number \[6.25\]. This can be done by using the factorization method. So, we will find the factors of the given number so that it satisfies the condition of square root \[x = y \times y = {y^2}\].

Complete step-by-step solution:
It is given that we need to find the square root of a decimal number \[6.25\]. As we know that a square root of a number is another number which when multiplied to itself gives the original number.
Normally a square root of a number can be found by using a long division method. But now we will find it by using the factorization method. Since the given number is a decimal number, we can write it as
\[6.25 = \dfrac{{625}}{{100}}\]
We need to find the square root of \[6.25\] that is \[\sqrt {6.25} \]. Taking square root on the above expression we get
\[\sqrt {6.25} = \sqrt {\dfrac{{625}}{{100}}} \]
The square root of a fraction can be separated into numerator and denominator.
\[\sqrt {{a^2}} = {\left( {{a^2}} \right)^{\dfrac{1}{2}}} = \left( {{a^{2 \times \dfrac{1}{2}}}} \right) = \left( {{a^1}} \right) = a\]
Now we need to find the factors of the numbers \[625\] and \[100\] in such a way that \[x = y \times y\].
The factors of the number \[625\] that is in the form \[x = y \times y\] are \[625 = 25 \times 25\]
Also, the factors of the number \[100\] that is the form \[x = y \times y\] are \[100 = 10 \times 10\]
Thus, we get \[\sqrt {6.25} = \dfrac{{\sqrt {25 \times 25} }}{{\sqrt {10 \times 10} }}\]
\[ \Rightarrow \sqrt {6.25} = \dfrac{{\sqrt {{{25}^2}} }}{{\sqrt {{{10}^2}} }}\]
\[ \Rightarrow \sqrt {6.25} = \dfrac{{25}}{{10}}\]
\[ \Rightarrow \sqrt {6.25} = 2.5\]
Thus, we have found that the square root of a number \[6.25\] is \[2.5\].

Note: The square root of the squared number equals the number alone. That is if \[{a^2}\] is a squared number then the square root of this squared number \[\sqrt {{a^2}} \] will be equal to \[a\]. This is because the square root of a number is nothing but the number raised to the power \[\dfrac{1}{2}\] thus we get \[\sqrt {{a^2}} = {\left( {{a^2}} \right)^{\dfrac{1}{2}}} = \left( {{a^{2 \times \dfrac{1}{2}}}} \right) = \left( {{a^1}} \right) = a\].