Rationalize the denominator and simplify if possible. √5+√3 / √5-√3
multiply top and bottom by the conjugate, and recall that (a-b)(a+b) = a^2-b^2
(√5+√3)/(√5-√3) * (√5+√3)/(√5+√3) = (√5+√3)^2/(5-3) = (8+2√15)/2 = 4+√15
To rationalize the denominator and simplify the given expression (√5 + √3) / (√5 - √3), you can use the conjugate method. The conjugate of a binomial expression is formed by changing the sign of the second term. In this case, the conjugate of (√5 - √3) is (√5 + √3).
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator (√5 + √3):
[ (√5 + √3) / (√5 - √3) ] * [ (√5 + √3) / (√5 + √3) ]
Expanding the numerator and denominator, we have:
[ (√5)(√5) + (√5)(√3) + (√3)(√5) + (√3)(√3) ] / [ (√5)(√5) + (√5)(√3) - (√3)(√5) - (√3)(√3) ]
This simplifies to:
[ 5 + 2√15 + 3 ] / [ 5 - 3 ]
Simplifying further:
[ 8 + 2√15 ] / 2
The final simplified expression is:
4 + √15