Question

What is the square root of $4.5$ ?

Answer 1

Hint: Here the above question is in the decimal form. And we have to find the square root of the decimal number. In algebra, a decimal number can be defined as a number whose whole number part and a fractional part is separated by a decimal point. And the dot in a decimal number is called a decimal point.

Complete step by step solution:
The given number is which we have to find the square root is:
$\Rightarrow \sqrt{4.5}$
In mathematics, a square root of a number x is a number y such that ${{y}^{2}}=x$, or we can say a number y whose square is x.
Since we can easily find the given number whose we have to find the square root is not a perfect square. So we have to simplify the given number to find out the square root.
Now the given number is $4.5$, we know that we can write any decimal number into fraction number so we will write the given number in fraction number as:
$\Rightarrow 4.5=\dfrac{45}{10}$
Now apply square root on both side of the above expression, then we get
$\Rightarrow \sqrt{4.5}=\sqrt{\dfrac{45}{10}}$
Now we know that if $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$ , so we will apply this then we get
$\Rightarrow \sqrt{4.5}=\dfrac{\sqrt{45}}{\sqrt{10}}$
Now we can write $45$ in the forms of prime factors as:
$\Rightarrow 45=3\cdot 3\cdot 5={{3}^{2}}\cdot 5$
Now putting this value in above expression, then we get
$\Rightarrow \sqrt{4.5}=\dfrac{\sqrt{{{3}^{2}}\cdot 5}}{10}$
We know ${{3}^{2}}$ is the perfect square so we can write
$\Rightarrow \sqrt{4.5}=\dfrac{3\sqrt{5}}{10}$
Now we will simplify it more, for this we will rationalize it, so multiply the numerator and denominator $\sqrt{10}$ , then we get
$\begin{align}
  & \Rightarrow \sqrt{4.5}=\dfrac{3\sqrt{5}}{\sqrt{10}}\times \dfrac{\sqrt{10}}{\sqrt{10}} \\
 & \Rightarrow \sqrt{4.5}=\dfrac{3\sqrt{50}}{10} \\
\end{align}$
After this we can write $50$ as $50={{5}^{2}}\cdot 2$ , since ${{5}^{2}}$ is a perfect square so we can rewrite expression as:
$\begin{align}
  & \Rightarrow \sqrt{4.5}=\dfrac{3\cdot 5\sqrt{2}}{10} \\
 & \Rightarrow \sqrt{4.5}=\dfrac{3\sqrt{2}}{2} \\
\end{align}$
Hence we get the required square root of $\sqrt{4.5}$ is $\dfrac{3\sqrt{2}}{2}$ .

Note: Here we used the simple way to solve this question, otherwise by calculating the square root for these types of numbers becomes lengthy. It is not hard to calculate the square root of perfect squares but if we do not have a perfect then always try to make that number in the simple form to reduce the mistakes.