Question

If $\sqrt{2}=1.4$and$\sqrt{3}=1.7$, find the value of each of the following, correct to one decimal place: $\dfrac{1}{3+2\sqrt{2}}$

Answer 1

Hint: Simplify the above equation using rationalisation and you will get simplified form and then calculation will become easier. Then insert the values to get to your answer and round off it to one decimal place.

Complete step by step answer:
An irrational number is a real number that cannot be expressed in the form of ratio, or we can say that, numbers that are not rational numbers are irrational numbers. They cannot be expressed in the form of a fraction, $p/q$form where p and q are integers and q is not equal to zero. Examples of irrational numbers are:$\sqrt{2},\sqrt{3},\pi $.....
Irrational numbers consist of non-terminating decimals.
When irrational numbers are present in numerator of fraction, calculation can be done easily as compared to when they are present in denominators of fractions, they make calculation more complicated. To avoid such complications, we use a method of rationalization.
Rationalisation can be defined as the process by which we eliminate radicals present in the denominators of the fraction.
$\dfrac{1}{3+2\sqrt{2}}$
Multiply numerator and denominator with$3-2\sqrt{2}$.
Now the equation will become:$\dfrac{3-2\sqrt{2}}{\left( 3+2\sqrt{2} \right)\left( 3-2\sqrt{2} \right)}$
Using identity $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$in the denominator we will get;
$\dfrac{3-2\sqrt{2}}{{{3}^{2}}-{{\left( 2\sqrt{2} \right)}^{2}}}$
Simplify the above equation,
$\dfrac{3-2\sqrt{2}}{\begin{align}
  & 9-8 \\
 & =3-2\sqrt{2} \\
\end{align}}$
Simplified equation is $3-2\sqrt{2}$
Insert the value of$\sqrt{2}$to get to your final answer.
$\begin{align}
  & 3-2\sqrt{2}=3-2\left( 1.4 \right) \\
 & =3-2.8 \\
 & =0.2 \\
\end{align}$
The answer of this question is $0.2$

Note: Rationalisation helps in simplifying the mathematical expression into a simpler form. It reduces the equation to more effective and simpler form. It simplifies fractions with roots in the denominator. If you try to solve the question without rationalisation that would be time consuming.