Question

rationalisethe denominator of (6 - 3sqrt(2))/(6 + 3sqrt(2))​

Answer 1

$\large{\underline{\pmb{\sf{Required\;solution...}}}}$

The question says that we need to rationalize the denominator of a fraction. The fraction is mentioned below:

$\pmb{ \sf{ \dfrac{6 - 3 \sqrt{2} }{6 + 3 \sqrt{2} } }}$

Explanation:

We nee to reduce the fraction as much as possible until the denominator gets rationalized.

How to rationalize the denominator? By reducing it! First, We will find the R.F of the denominator and multiply it with it! Then, we will use laws of exponents only if needed and solve it until the denominator is rationalized!

Step by step explanation:

$\hookrightarrow \: \pmb{ \sf{ \dfrac{6 - 3 \sqrt{2} }{6 + 3 \sqrt{2} } }}$

R.F of 6 + 3√2 is 6 - 3√2,

$\hookrightarrow \: \pmb{ \sf{ \dfrac{6 - 3 \sqrt{2} }{6 + 3 \sqrt{2} } \times \dfrac{6 - 3 \sqrt{2} }{6 - 3 \sqrt{2} } }}$

Multiplying,

$\hookrightarrow \: \pmb{ \sf{ \dfrac{ {(6 - 3 \sqrt{2})}^{2} }{ {(6)}^{2} - ({3 \sqrt{2})}^{2} } { } }}$

$\hookrightarrow \: \pmb{ \sf{ \dfrac{ {36 +18 - 36 \sqrt{2}} }{ {(6)}^{2} - ({3 \sqrt{2})}^{2} } { } }}$

$\hookrightarrow \: \pmb{ \sf{ \dfrac{ {36 +18 - 36 \sqrt{2}} }{ 36 - 18 } { } }}$

$\hookrightarrow \: \pmb{ \sf{ \dfrac{ {36 +18 - 36 \sqrt{2}} }{ 18 } { } }}$

$\hookrightarrow \: \pmb{ \sf{ \dfrac{ {54 - 36 \sqrt{2}} }{ 18 } { } }}$

$\hookrightarrow \: \pmb{ \sf{ \dfrac{ {18(3 - 2 \sqrt{2})} }{ 18 } { } }}$

$\hookrightarrow \: \pmb{ \sf{ (3 - 2 \sqrt{2})} }$

Henceforth, Denominator of the fraction is rationalized.

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Additional information:

Rational numbers: The numbers which can be written in p/q form. where q ≠ 0 i.e., q is not equal to zero. For example - 7/8, 9/8, 567/738.

Irrational numbers: Irrational numbers are the totally opposite to rational numbers. They cannot be expressed in the form of p/q. The best examples for irrational numbers is e and π.