Answer :
Answer:
C) [tex]-3-2\sqrt{2}[/tex].
Step-by-step explanation:
Rationalizing Fractions
When a fraction has an irrational number or, in our case, a radical term(s) in its denominator, its conjugate must be multiplied to the top and bottom of the fraction to transform the fraction to have a rational value.
Conjugate
The conjugate is simply the expression, a binomial, with the sign next to the irrational term flipped. In cases where the binominal has two radical terms, the convention is to flip the sign of the second radical.
Rationalizing the Problem
We're given the fraction,
[tex]\dfrac{\sqrt{5}+\sqrt{10}}{\sqrt{5}-\sqrt{10}}[/tex].
We first identify the conjugate of the denominator before multiplying it to numerator and denominator and further simplifying it.
The conjugate of [tex]\sqrt{5} -\sqrt{10}[/tex], using the rule for denominator with two radicals, is [tex]\sqrt{5} +\sqrt{10}[/tex].
Then rationalizing our fraction we get,
[tex]\dfrac{\sqrt{5}+\sqrt{10}}{\sqrt{5}-\sqrt{10}}\: \cdot\dfrac{\sqrt{5}+\sqrt{10}}{\sqrt{5}+\sqrt{10}}[/tex].
Noticing the terms being multiplied in the denominator, we see it follows the differences of squares formula: [tex]a^2-b^2=(a+b)(a-b)[/tex].
So, the denominator's new value is,
[tex](\sqrt{5})^2 -(\sqrt{10})^2=5-10=-5[/tex].
Focusing on the numerator, we see that the same binominal is being multiplied to itself, thus following the [tex](a+b)^2[/tex] formula:[tex](a+b)^2=(a+b)(a+b)=a^2+2ab+b^2[/tex].
So, the numerator's new value is,
[tex](\sqrt{5})^2+2(\sqrt{5} )(\sqrt{10} ) +(\sqrt{10} )^2[/tex]
[tex]=5+2\sqrt{5\cdot10} +10\\\\=15 + 2\sqrt{50}\\\\=15+ 2(5\sqrt{2})\\\\\Longrightarrow15+10\sqrt{2}[/tex].
Our final answer is,
[tex]\dfrac{15+10\sqrt{2}}{-5} =\dfrac{15}{-5} +\dfrac{10\sqrt{2}}{-5} =\boxed{-3-2\sqrt{2}}[/tex].
