Rationalize the Denominator.
5/(3+sqrt(5))
To rationalize the denominator of the expression 5/(3+√5), you can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 3+√5 is 3-√5.
Here's how you can do it step by step:
1. Start with the expression 5/(3+√5).
2. Multiply the numerator and denominator by the conjugate of the denominator, which is 3-√5:
(5/(3+√5)) * ((3-√5)/(3-√5))
3. Simplify the expression by using the distributive property:
(5 * (3-√5)) / ((3+√5) * (3-√5))
4. Apply the FOIL method to multiply the denominators:
(15-√25) / (9-√15+√15-√25)
5. Simplify the expression:
(15-√25) / (9-√25)
6. The square root of 25 is 5, so simplify further:
(15-5) / (9-5)
7. Simplify the expression:
10 / 4
8. Reduce the fraction to its simplest form:
5 / 2
So, the rationalized form of 5/(3+√5) is 5/2.