Question

How do you simplify $-\sqrt{144}$?

Answer 1

Hint: We try to form the indices formula for the value $n=2$ of \[{{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}\]. This is a square root of 144. We find the prime factorisation of 144. Then we take one digit out of the two same number of primes. 144 is square of 12. Therefore, the square root of $144$ gives the result of 12.

Complete step by step solution:
We need to find the value of the algebraic form of $-\sqrt{144}$. This is a square root form. The sign of the square root will be negative.
The given value is the form of indices. We are trying to find the root value of 144.
We know the theorem of indices \[{{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}\]. Putting value 2 we get \[{{a}^{\dfrac{1}{2}}}=\sqrt{a}\].
We need to find the prime factorisation of the given number 144.
$\begin{align}
  & 2\left| \!{\underline {\,
  144 \,}} \right. \\
 & 2\left| \!{\underline {\,
  72 \,}} \right. \\
 & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Therefore, $144=2\times 2\times 2\times 2\times 3\times 3$.
For finding the square root, we need to take one digit out of the two same number of primes.
This means in the root value of $144=2\times 2\times 2\times 2\times 3\times 3$, we will take out two 2s and one 3.
So, $\sqrt{144}=\sqrt{2\times 2\times 2\times 2\times 3\times 3}=2\times 2\times 3=12$. Basically 144 is the multiplication of two sixes.
Therefore, $-\sqrt{144}=-12$.
We can also use the theorem of indices \[{{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}\]. We know that $144={{12}^{2}}$.
We need to find $-\sqrt{144}$ which gives $-{{\left( 144 \right)}^{\dfrac{1}{2}}}=-{{\left( {{12}^{2}} \right)}^{\dfrac{1}{2}}}=-\left( {{12}^{2\times \dfrac{1}{2}}} \right)=-12$.
Therefore, the value of $-\sqrt{144}$ is $-12$.

Note: We can also use the variable form where we can take $x=-\sqrt{144}$. But we need to remember that we can’t use the square on both sides of the equation $x=-\sqrt{144}$ as in that case we are taking an extra value of negative as in $-12$ as a root value. Then this linear equation becomes a quadratic equation.