Question

How do you simplify \[{{144}^{\dfrac{1}{2}}}\]?

Answer 1

Hint: To solve the given question we will need the following properties. The property of exponents which states that \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\] here, \[a,m,n\in \]Real numbers. We should also know that if a is the square of b, then it can be written as \[a={{b}^{2}}\], we will use these properties to solve the given question.

Complete step by step answer:
We are asked to simplify \[{{144}^{\dfrac{1}{2}}}\], which means we have to find its value. We know that 144 is the square 12. Using the property which states that, if a is the square of b, then it can be written as \[a={{b}^{2}}\]. Here we have a = 144, and b = 12. By substituting the value, it can be written as, \[144={{12}^{2}}\]. Using this in the given question, it can be simplified as,
\[\begin{align}
  & \Rightarrow {{144}^{\dfrac{1}{2}}} \\
 & \Rightarrow {{\left( {{12}^{2}} \right)}^{\dfrac{1}{2}}} \\
\end{align}\]
We know the property of exponents which states that, \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\] here, \[a,m,n\in \] Real numbers. We have a = 144, m = 2, and n = \[\dfrac{1}{2}\]. Using this property in the above expression it becomes,
\[\begin{align}
  & \Rightarrow {{\left( {{12}^{2}} \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow {{12}^{2\times \dfrac{1}{2}}} \\
 & \Rightarrow {{12}^{1}}=12 \\
\end{align}\]
Hence the given expression \[{{144}^{\dfrac{1}{2}}}\] can be written in simplified form as 12.

Note:
 These types of questions can be solved by remembering the values of squares, cubes, square roots, and cube roots of the numbers. We can also solve this question by factorization the number 144, and then taking the factors out of the square root which appears twice. This can be done as follows, we know that \[{{144}^{\dfrac{1}{2}}}\] can also be written as \[\sqrt{144}\]. Hence, \[\sqrt{144}=\sqrt{2\times 2\times 2\times 2\times 3\times 3}=2\times 2\times 3=12\]. We can see that we are getting the same answer from both methods.