Question

Express
$\frac{4}{2 \sqrt{5} - 3 } \times \frac{ \sqrt{180} - \sqrt{27} } {( \sqrt{2} - \sqrt{5}) {}^{2} } \times \sqrt{1 \times \frac{9}{16} }$
In its simplest form
$\frac{a}{b} (c + d \sqrt{e} )$
where a, b, c , d and e are integers.
PLEASE GIVE A STEP-BY-STEP SOLUTION! Thank you!​

Answer 1

Answer:

$\frac{15}{9} (7 + 2 \sqrt{10} )$

Step-by-step explanation:

The solution is in the picture

At line 7, the denominator of the expression contains irrational number (which is surd 10). This shows that the expression is not simplified

At line 8 of my solution, multiply both numerator and denominator by the conjugate of

$7 - 2 \sqrt{10}$

which is

$7 + 2 \sqrt{10}$

in order to eliminate irrational number at denominator

More info of conjugate

The conjugate of a+b is a-b,

Why conjugate? Try it yourself!

Hint: (a+b)(a-b) = a^2 - b^2