Question

rationalize the denominator of 4 by root 11 minus 7​

Answer 1

Question :-

Rationalise the denominator of $\tt \dfrac{4}{ \sqrt{11} - 7}$

Answer :-

$\tt - \dfrac{2 \sqrt{11} + 14}{19}$

Solution :-

$\tt \dfrac{4}{ \sqrt{11} - 7}$

The rationalising factor of √11 - 7 is √11 + 7 . So multiply both numerator and denominator with rationalising factor.

$\tt = \dfrac{4}{ \sqrt{11} - 7} \times \dfrac{ \sqrt{11} + 7 }{ \sqrt{11} + 7}$

$\tt = \dfrac{4( \sqrt{11} + 7) }{ (\sqrt{11} - 7)( \sqrt{11} + 7) }$

$\tt = \dfrac{4 \sqrt{11} + 28 }{ (\sqrt{11} - 7)( \sqrt{11} + 7) }$

$\tt = \dfrac{4 \sqrt{11} + 28 }{ {( \sqrt{11}) }^{2} - {(7)}^{2} }$

[Since (x + y)(x - y) = x² - y² and above x = √11 and y = 7]

$\tt = \dfrac{4 \sqrt{11} + 28 }{11 - 49}$

$\tt = \dfrac{4 \sqrt{11} + 28 }{ - 38}$

It can be written as

$\tt = \dfrac{ \cancel 2(2 \sqrt{11} + 14)}{ \cancel 2( - 19)}$

$\tt = \dfrac{2 \sqrt{11} + 14}{ - 19}$

$\tt = - \dfrac{2 \sqrt{11} + 14}{19}$