Question

How do you simplify \[\sqrt{9{{x}^{2}}}\]?

Answer 1

Hint: In this problem, we have to simplify \[\sqrt{9{{x}^{2}}}\]. We can write in whole square form inside the square root, then we can convert the square root, that is radical form, to the fractional exponent. We can then cancel the power of the terms and the fractional exponent raised to get the simplified form. We can use the formula \[\sqrt[n]{a}={{a}^{\dfrac{1}{n}}}\] to convert the radical to fractional exponent.

Complete step by step answer:
We know that the given expression to be simplified is \[\sqrt{9{{x}^{2}}}\].
We can write the terms inside the square root in whole square form, as we know that 3 squares are 9.
\[\Rightarrow \sqrt{9{{x}^{2}}}=\sqrt{{{\left( 3x \right)}^{2}}}\] …… (1)
Now we can convert the square root form, that is the radical form, into fractional exponent form.
The formula to convert the radical form to fractional exponent form is \[\sqrt[n]{a}={{a}^{\dfrac{1}{n}}}\].
Now we can apply the above formula in the expression (1), we get
\[\Rightarrow \sqrt[2]{{{\left( 3x \right)}^{2}}}={{\left( 3x \right)}^{2\times \dfrac{1}{2}}}\]
Now we can cancel the similar numbers in the power, we get
\[\Rightarrow 3x\]
Therefore, the simplified form of \[\sqrt{9{{x}^{2}}}\]is \[3x\].

Note:
Students make mistakes while writing the whole square form inside the square root, it is possible only when the given numbers are perfect squares. If the given numbers are perfect squares, we can solve it in a simple way, if it is not a perfect square, then we can use another method to solve those problems. We should remember that, to solve these types of problems, we should know how to convert the radicals to fractional exponents.