How do you rationalise the denominator of #(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))# and express in the form #a+bsqrt(3)# ?

Answer 1

#(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3)) = 4+sqrt(15)#

You could have #a=4# and #b=sqrt(5)# but #sqrt(5)# is irrational.

To rationalise the denominator, multiply both numerator and denominator by the "rational conjugate" #(sqrt(5)+sqrt(3))#, then simplify...
#(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3)) = (sqrt(5)+sqrt(3))^2/((sqrt(5)-sqrt(3))(sqrt(5)+sqrt(3)))#
#color(white)((sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))) = ((sqrt(5))^2+2sqrt(5)sqrt(3)+(sqrt(3))^2)/((sqrt(5))^2-(sqrt(3))^2)#
#color(white)((sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))) = (5+2sqrt(15)+3)/(5-3)#
#color(white)((sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))) = 4+sqrt(15)#
Note that #sqrt(15)=sqrt(3)sqrt(5)# is an irrational multiple of #sqrt(3)#, not a rational multiple.
The result is not expressible in the form #a+bsqrt(3)# for rational #a, b#.
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Answer 2

To rationalize the denominator of the expression (\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}), you multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of (\sqrt{5} - \sqrt{3}) is (\sqrt{5} + \sqrt{3}).

So, multiplying both the numerator and denominator by (\sqrt{5} + \sqrt{3}), we get:

[\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} = \frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5})^2 - (\sqrt{3})^2}]

Expanding the numerator and denominator:

[= \frac{5 + 2\sqrt{15} + 3}{5 - 3} = \frac{8 + 2\sqrt{15}}{2} = 4 + \sqrt{15}]

So, (\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}) when rationalized becomes (4 + \sqrt{15}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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