Question

rationalise the denominator 1/5+√6​

Answer 1

To rationalise the denominator of :

$\longrightarrow \sf { \dfrac{1}{5+ \sqrt{6} } }$

Answer :

$\longrightarrow \sf { \dfrac{5-\sqrt{6}}{19 } }$

Solution :

Given fraction,

$\longrightarrow \sf { \dfrac{1}{5+ \sqrt{6} } }$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

We are asked to rationalise the denominator of this fraction.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

In order to rationalise the denominator of any fraction, we need to multiply the the rationalising factor of the denominator with both the numerator and the denominator of the given fraction.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Here,

$\longrightarrow \sf{ Denominator = 5+ \sqrt{6} }$

Rationalising factor of (a + b) is (a – b), therefore rationalising factor of $\sf{ (5+ \sqrt{6} )}$ is $\sf{ (5- \sqrt{6} )}$. So, we'll multiply $\sf{ (5- \sqrt{6} )}$ with both numerator and denominator of the given fraction.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow \sf { \dfrac{1}{5+ \sqrt{6} } \times \dfrac{5- \sqrt{6}}{5- \sqrt{6}} }$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow \sf { \dfrac{1(5- \sqrt{6})}{(5+ \sqrt{6})(5- \sqrt{6}) } }$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

By using identity,

• $\bf { (a+b)(a-b) = a^2 - b^2}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow \sf { \dfrac{5- \sqrt{6} }{(5)^2 - (\sqrt{6})^2 } }$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow \sf { \dfrac{5- \sqrt{6} }{25- 6 } }$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow\boxed{ \sf { \dfrac{5- \sqrt{6} }{19}} }$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Hence, rationalised!

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

More Information :

• (√a)² = a

• √a√b = √ab

• √a/√b = √a/b

• (√a + √b)(√a - √b) = a - b

• (a + √b)(a - √b) = a² - b

• (√a ± √b)² = a ± 2√ab + b

• (√a + √b)(√c + √d) = √ac + √ad + √bc + √bd