Question

How do you simplify ($4$ sqrt $45$)/ ($5$ sqrt $8$)?

Answer 1

Hint: To solve the given expression, we must express it mathematically. It will be mathematically expressed as $\dfrac{4\sqrt{45}}{5\sqrt{8}}$. Since the denominator in this expression is irrational, so for simplifying this expression, we first have to rationalise the denominator. For this we have to multiply and divide the expression by $\sqrt{8}$ and then simplify the radicals on the numerator to get the final simplified expression.

Complete step by step solution:
The given expression in the above question can be expressed mathematically as
$\Rightarrow E=\dfrac{4\sqrt{45}}{5\sqrt{8}}$
We can observe in the above expression that the denominator is irrational. So the denominator has to be rationalised first. For the we multiply and divide the above expression by $\sqrt{8}$ to get
\[\begin{align}
  & \Rightarrow E=\dfrac{4\sqrt{45}}{5\sqrt{8}}\times \dfrac{\sqrt{8}}{\sqrt{8}} \\
 & \Rightarrow E=\dfrac{4\sqrt{45}\times \sqrt{8}}{5\sqrt{8}\times \sqrt{8}} \\
 & \Rightarrow E=\dfrac{4\sqrt{45}\times \sqrt{8}}{5{{\sqrt{8}}^{2}}} \\
 & \Rightarrow E=\dfrac{4\sqrt{45}\times \sqrt{8}}{5\times 8} \\
 & \Rightarrow E=\dfrac{4\sqrt{45}\times \sqrt{8}}{40} \\
\end{align}\]
Now, from the laws of radicals, we know that $\sqrt{a}\sqrt{b}=\sqrt{ab}$. Therefore, the above expression becomes
\[\begin{align}
  & \Rightarrow E=\dfrac{4\sqrt{45\times 8}}{40} \\
 & \Rightarrow E=\dfrac{4\sqrt{360}}{40} \\
\end{align}\]
Now, separating the highest perfect square from the radical, we get
\[\Rightarrow E=\dfrac{4\sqrt{36\times 10}}{40}\]
Using the rule $\sqrt{ab}=\sqrt{a}\sqrt{b}$, we get
\[\Rightarrow E=\dfrac{4\sqrt{36}\sqrt{10}}{40}\]
Now, we know that $\sqrt{36}=6$. Putting this above we get
\[\begin{align}
  & \Rightarrow E=\dfrac{4\times 6\sqrt{10}}{40} \\
 & \Rightarrow E=\dfrac{24\sqrt{10}}{40} \\
\end{align}\]
Simplifying the numerator and the denominator by cancelling the common factors, we get
\[\Rightarrow E=\dfrac{3\sqrt{10}}{5}\]
Hence, the given expression is simplified.

Note:
Make sure that in the final simplified expression, the denominator is rational. Before rationalising the denominator, we can also simplify the radicals on the numerator and the denominator as $\sqrt{45}=3\sqrt{5}$ and $\sqrt{8}=2\sqrt{2}$ and then do the rationalization of the denominator. Also, for the rationalization of the denominator, we do not need to multiply and divide the expression by the entire denominator. Only the irrational term is to be multiplied and divided.