Question

What is the square root of 1200 simplified in radical form?

Answer 1

Hint: We try to form the indices formula for the value 2. This is a square root of 1200. We find the prime factorisation of 1200. Then we take one digit out of the two same number of primes. There will be 3 remaining in the root which can’t be taken out. We keep the cube root in its simplest form.

Complete step by step solution:
We need to find the value of the algebraic form of $\sqrt[2]{1200}$. This is a square root form.
The given value is the form of indices. We are trying to find the root value of 1200.
We know the theorem of indices \[{{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}\]. Putting value 3 we get \[{{a}^{\dfrac{1}{2}}}=\sqrt[2]{a}\].
We need to find the prime factorisation of the given number 1200.
$\begin{align}
  & 2\left| \!{\underline {\,
  1200 \,}} \right. \\
 & 2\left| \!{\underline {\,
  600 \,}} \right. \\
 & 2\left| \!{\underline {\,
  300 \,}} \right. \\
 & 2\left| \!{\underline {\,
  150 \,}} \right. \\
 & 3\left| \!{\underline {\,
  75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Therefore, \[1200=2\times 2\times 2\times 2\times 3\times 5\times 5\].
For finding the square root, we need to take one digit out of the two same number of primes.
This means in the square root value of \[1200=2\times 2\times 2\times 2\times 3\times 5\times 5\], we will take out one 5 and two 2s from the multiplication.
So, $\sqrt[2]{1200}=\sqrt[2]{2\times 2\times 2\times 2\times 3\times 5\times 5}=2\times 2\times 5\sqrt{3}=20\sqrt{3}$.
We can also use the theorem of indices \[{{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}\]. We know that $1200={{\left( 20\sqrt{3} \right)}^{2}}$.
We need to find $\sqrt[2]{1200}$ which gives \[\sqrt{1200}=\sqrt{{{\left( 20\sqrt{3} \right)}^{2}}}={{\left[ {{\left( 20\sqrt{3} \right)}^{2}} \right]}^{\dfrac{1}{2}}}={{\left( 20\sqrt{3} \right)}^{2\times \dfrac{1}{2}}}=20\sqrt{3}\].
Therefore, the value of $\sqrt[2]{1200}$ is $20\sqrt{3}$.

Note: We can also use the variable form where we can take $x=\sqrt[2]{1200}$. But we need to remember that we can’t use the cube on both sides of the equation $x=\sqrt[2]{1200}$ as in that case we are taking two extra values as a root value. Then this linear equation becomes a cubic equation