Question

How do you simplify Square root of 63 and square root of 28.

Answer 1

Hint: In the problem we can see that we need to simplify the given integer and find the square root of the given integer. For this we need to factorize the given integer and by using the exponential rules we will write the given integer in the form of exponentials of its factors. Now we will apply the square root function as the half power of the integer. Now We will get the simplified solution. We know that factorization is the process of writing an integer in the form of multiplication of all its factors. We will first divide the given integer with $2$ and check if it gives zero remainder or not, if it gives zero remainder, we will write the given integer as the product of the $2$ and the quotient. If it is not given zero remainder, we will check for the number which gives zero remainder by dividing the given integer with the next integers which are odd like $3,5,7,11,13,...$. After that we will do the same process for the quotients and write the given integer in the form of multiplication of all the factors.

Complete step-by-step solution:
Given integers $63$, $28$.
Let us consider the integer $63$.
Checking whether $63$ is divisible by $2$ or not. We can clearly see that $63$ is not divisible by $2$, as it gives a remainder $1$ when we divide $63$ with $2$.
Checking whether $63$ is divisible by $3$ or not. We can clearly see that $63$ is divisible by $3$ and it gives $21$ as quotient. Now we can write the integer $63$ as
$63=3\times 21$.
Now we will factorize the number $21$.
Checking whether $21$ is divisible by $2$ or not. We can clearly see that $21$ is not divisible by $2$, as it gives a remainder $1$ when we divide $21$ with $2$.
Checking whether $21$ is divisible by $3$ or not. We can clearly see that $21$ is divisible by $3$ and it gives $7$ as quotient. Now we can write the integer $21$ as
$21=3\times 7$
From the above value, we can write $63$ as
$\begin{align}
  & 63=3\times 21 \\
 & \Rightarrow 63=3\times 3\times 7 \\
\end{align}$
We have the exponential rule $a\times a={{a}^{2}}$, then we will get
$63={{3}^{2}}\times 7$
Now apply square root function on both sides of the above equation, then we will get
$\sqrt{63}=\sqrt{{{3}^{2}}\times 7}$
We have the square root function as half power of the variables, i.e. $\sqrt{a}={{a}^{\dfrac{1}{2}}}$. Then
$\begin{align}
  & \sqrt{63}={{\left( 63 \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{63}={{\left( {{3}^{2}}\times 7 \right)}^{\dfrac{1}{2}}} \\
\end{align}$
We have the exponential rule ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$, then we will get
$\sqrt{63}={{\left( {{3}^{2}} \right)}^{\dfrac{1}{2}}}\times {{7}^{\dfrac{1}{2}}}$
We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$, then we will get
$\begin{align}
  & \sqrt{63}={{3}^{2\times \dfrac{1}{2}}}\times \sqrt{7} \\
 & \Rightarrow \sqrt{63}=3\sqrt{7} \\
\end{align}$
Hence the simplified form of $\sqrt{63}$ is $3\sqrt{7}$.
Let us consider the integer $28$.
Checking whether $28$ is divisible by $2$ or not. We can clearly see that $28$ is divisible by $2$ and it gives $14$ as quotient. Now we can write the integer $28$ as
$28=2\times 14$
Now we will factorize the number $14$.
Checking whether $14$ is divisible by $2$ or not. We can clearly see that $14$ is divisible by $2$ and it gives $7$ as quotient. Now we can write the integer $14$ as
$14=2\times 7$
From the above value, we can write $28$ as
$\begin{align}
  & 28=2\times 14 \\
 & \Rightarrow 28=2\times 2\times 7 \\
\end{align}$
We have the exponential rule $a\times a={{a}^{2}}$, then we will get
$28={{2}^{2}}\times 7$
Now apply square root function on both sides of the above equation, then we will get
$\sqrt{28}=\sqrt{{{2}^{2}}\times 7}$
We have the square root function as half power of the variables, i.e. $\sqrt{a}={{a}^{\dfrac{1}{2}}}$. Then
$\begin{align}
  & \sqrt{28}={{\left( 28 \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{28}={{\left( {{2}^{2}}\times 7 \right)}^{\dfrac{1}{2}}} \\
\end{align}$
We have the exponential rule ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$, then we will get
$\sqrt{28}={{\left( {{2}^{2}} \right)}^{\dfrac{1}{2}}}\times {{7}^{\dfrac{1}{2}}}$
We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$, then we will get
$\begin{align}
  & \sqrt{28}={{2}^{2\times \dfrac{1}{2}}}\times \sqrt{7} \\
 & \Rightarrow \sqrt{28}=2\sqrt{7} \\
\end{align}$
Hence the simplified form of $\sqrt{28}$ is $2\sqrt{7}$.

Note: For doing factorization one should clearly know the divisions and the terminology like remainder and quotient. It is better to know the division rules to speed up the process. We have division rules for the integers $2,3,4,5,6,10,11$. These are the basic factors for any integer, so knowing the division rules of these numbers will be an advantage.