Question

How do you simplify $\dfrac{1}{{\sqrt {32} }}$?

Answer 1

Hint: This problem deals with simplifying the given radical expression. This is done by simplifying the radical expressions using the product and quotient rule for radicals. Use formulas involving radicals. Evaluate given square root and cube root functions. An algebraic expression that contains radicals is called a radical expression. We use the product and quotient rules to simplify them.

Complete step-by-step answer:
Given the radical expression $\dfrac{1}{{\sqrt {32} }}$, consider this expression as given below:
$ \Rightarrow \dfrac{1}{{\sqrt {32} }}$
Now rationalize the denominator by multiplying the given fraction with $\sqrt {32} $. That is multiplying and dividing the given fraction with $\sqrt {32} $, as shown below:
$ \Rightarrow \dfrac{1}{{\sqrt {32} }} \times \dfrac{{\sqrt {32} }}{{\sqrt {32} }}$
$ \Rightarrow \dfrac{{\sqrt {32} }}{{32}}$
Now simplifying the numerator which is $\sqrt {32} $ , as given below:
The $\sqrt {32} $ can be written as: $\sqrt {32} = \sqrt {8 \times 4} $
$ \Rightarrow \sqrt {32} = \sqrt {2 \times 2 \times 2 \times 2 \times 2} $
We know that 8, is the cube of 2, and whereas 4, is the square of 2.
$\therefore \sqrt {32} = 4\sqrt 2 $
Now substituting the value of $\sqrt {32} = 4\sqrt 2 $ in the numerator, as shown below:
$ \Rightarrow \dfrac{{\sqrt {32} }}{{32}} = \dfrac{{4\sqrt 2 }}{{32}}$
Now simplifying the numbers, in the numerator and denominator, which are outside the root, as shown below:
$ \Rightarrow \dfrac{{4\sqrt 2 }}{{32}} = \dfrac{{\sqrt 2 }}{8}$
So the value of the radical expression $\dfrac{1}{{\sqrt {32} }}$, is given by:
$\therefore \dfrac{1}{{\sqrt {32} }} = \dfrac{1}{8}\sqrt 2 $

Final Answer: The simplification of the radical expression of $\dfrac{1}{{\sqrt {32} }}$ is equal to $\dfrac{{\sqrt 2 }}{8}$.

Note:
Please note that if you want to multiply, first coefficients are multiplied with each other and the sub-radical amounts each other, placing the latter product under the radical sign common and the result is simplified.
But if you want to divide, then the coefficients are divided among themselves and sub-radical amounts each other, placing the latter quotient under the radical common and the result is simplified.