Question

rationalize the denominator

1/3-√2​

Answer 1

Step-by-step explanation:

Sorry my hand writing is not so good

Answer 2

AnswEr:-  

Your Answer Is $\tt\dfrac{3+\sqrt{2}}{7}.$

 ExplanaTion:-

Given number:-

• $\tt\dfrac{1}{3-\sqrt{2}}.$

To Rationalize:-

• The denominator of the given number.

 ID Used:-

$\tt\longrightarrow(a+b)(a-b) = a^2-b^2.$

Concept Used:-

$\tt\dfrac{1}{a}\:\:can\:\:be\:\:written\:\:as,\\\\ \tt\dfrac{1}{a}\times\frac{a}{a}\:\:because\\\\ \tt \frac{a}{a}=1 \\\\ and\:\:\frac{1}{a}\times1=\frac{1}{a}.$

Now,

Before Rationalizing let us first understand the meaning of rationalization:-  

$\rm\underline{Rationalization:-}$Converting an irrational number to a rational number is called Rationalizing, and here rationalizing the denominator means we have convert denominator (Which is an irrational number) to a rational number.  

So,

Lets rationalize the denominator of give number:-   

$\tt\mapsto\dfrac{1}{3-\sqrt{2}}. \\ \\ \tt=\frac{1}{3-\sqrt{2}}\times\frac{3+\sqrt{2}}{3+\sqrt{2}}.\\ \\ \rm[By\:\:Using\:\:The\:\:Concept].\\ \\ \tt=\dfrac{1(3+\sqrt{2})}{(3+\sqrt{2})(3-\sqrt{2})}.\\ \\ \tt=\frac{1}{(3)^2-(\sqrt{2})^2}.\\\\ \rm[By\:\:Using\:\:The\:\:ID].\\\\ \tt = \frac{3+\sqrt{2}}{(3\times3)(\sqrt{2}\times\sqrt{2})}.\\\\ \tt = \frac{3+\sqrt{2}}{9-2}.\\\\ \tt =\frac{3+\sqrt{2}}{7}.$

Clearly we can see that the denominator is a rational number.

Hence the denominator is rationalized.