Question

What is the square root of 108 in simplest radical form?

Answer 1

Hint: Now to find the simplest radical form we will first factorize the number 108. Now with the help of factors we will try to take the square numbers out of the roots. Hence we will get the simplest radical form of the given expression.

Complete step-by-step solution:
Now first let us understand the concept of square and square root.
Now we know that when a number is raised to another number we get exponents. Now the number which is raised is called power. The power denotes the number of time the base number must be multiplied to itself.
Now let us understand this with an example. Suppose we have ${{4}^{3}}$ now since the power here is 3 we need to multiply 4 to itself 3 times. Hence we get, ${{4}^{3}}=4\times 4\times 4=64$ .
Now if the power of any number is 2 then it is called the square of that number. Hence if we have ${{3}^{2}}=9$ we say that 9 is square of 3.
Now square root is nothing but the inverse of square. Hence if 9 is square of 3 then we can say that 3 is square root of 9. Square root of any number x is denoted by $\sqrt{x}$ .
Now we want to find the square root of $108$ .
First let us prime factorize 108.
We have $108=2\times 2\times 3\times 3\times 3$
Hence we get,
$\begin{align}
  & \Rightarrow \sqrt{108}=\sqrt{2\times 2\times 3\times 3\times 3} \\
 & \Rightarrow \sqrt{108}=\sqrt{6\times 6\times 3} \\
 & \Rightarrow \sqrt{108}=\sqrt{{{6}^{2}}\times 3} \\
\end{align}$
Now since square is nothing but inverse of square we can take 6 out of the root. Hence we get,
$\Rightarrow \sqrt{108}=6\sqrt{3}$ .
Hence we get the solution to the given equation is $6\sqrt{3}$ .

Note: Now note that similar to square if we have 3 in the power then the function is called a cube of numbers. Hence cube of number x will be ${{x}^{3}}$ . Hence the inverse of cube will be cube root. The cube root of is represented by $\sqrt[3]{x}$ .