Question

How do you simplify 4 times the square root of 8 over the square root of 2?

Answer 1

Hint: First try to convert the word problem to numeric expression. After the expression is formed since $\sqrt{2}$ is present in the denominator, hence simplify it by breaking $\sqrt{8}$ as $2\sqrt{2}$. So that the $\sqrt{2}$ part will be cancelled out both from numerator and denominator to give the simplified value.

Complete step-by-step answer:
Let’s form the expression first
4 times the square root of 8 $=4\cdot \sqrt{8}$
The square root of 2 $=\sqrt{2}$
So, 4 times the square root of 8 over the square root of 2 $=\dfrac{4\cdot \sqrt{8}}{\sqrt{2}}$
Hence, $\dfrac{4\cdot \sqrt{8}}{\sqrt{2}}$ is our required expression.
For simplification, simplifying $\sqrt{8}$ part first
$\sqrt{8}$ can be written as $\sqrt{8}=\sqrt{2\times 2\times 2}=\sqrt{2}\cdot \sqrt{2}\cdot \sqrt{2}=\left( \sqrt{2}\cdot \sqrt{2} \right)\cdot \sqrt{2}=2\sqrt{2}$
Putting the value of $\sqrt{8}$ in the expression, we get
$=\dfrac{4\cdot 2\sqrt{2}}{\sqrt{2}}$
Cancelling out $\sqrt{2}$ both from numerator and denominator, we get
$\begin{align}
  & =4\cdot 2 \\
 & =8 \\
\end{align}$
So, $\dfrac{4\cdot \sqrt{8}}{\sqrt{2}}=8$
This is the simplified value of the given expression.

Note: Converting the word problem to proper numeric expression should be the first approach to solve such a question. This can be done by taking and understanding each part of the given question. For the simplification part the terms of the expressions should be factored so that they can be cancelled out from numerator and denominator to give the simplified value. The above expression can also be solved as
$\dfrac{4\cdot \sqrt{8}}{\sqrt{2}}$
Multiplying $\sqrt{2}$ in both numerator and denominator, we get
$=\dfrac{4\cdot \sqrt{8}\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}$
$\sqrt{8}\cdot \sqrt{2}$ can be written as $\sqrt{8}\cdot \sqrt{2}=\sqrt{8\cdot 2}=\sqrt{16}=4$
$\sqrt{2}\cdot \sqrt{2}$ can be written as $\sqrt{2}\cdot \sqrt{2}=\sqrt{2.2}=\sqrt{4}=2$
Now, the expression is
$\begin{align}
  & =\dfrac{4\cdot 4}{2} \\
 & =\dfrac{16}{2} \\
 & =8 \\
\end{align}$
This is another method for solving such questions.