Rationalize the denominator using the conjugate.
1/1-√7+√5
please show all the steps and explain it
i don't get it
To rationalize the denominator using the conjugate, we need to get rid of the square roots in the denominator. The conjugate of a binomial expression is formed by changing the sign between the terms. In this case, the conjugate of 1-√7+√5 is 1+√7-√5.
Here are the steps to rationalize the given expression:
Step 1: Start with the given expression 1/(1-√7+√5).
Step 2: Multiply the numerator and denominator by the conjugate of the denominator, which is 1+√7-√5. This will ensure that we eliminate the square roots in the denominator.
(1/(1-√7+√5)) * ((1+√7-√5)/(1+√7-√5))
Step 3: Simplify the resulting expression. We can use the distributive property to do this:
(1*(1+√7-√5))/((1-√7+√5)*(1+√7-√5))
= (1+√7-√5)/(1 - (√7-√5) + (√7-√5) - (√7+√5))
= (1+√7-√5)/(1 - 2√7 + 2√5)
Step 4: Further simplify the denominator:
= (1+√7-√5)/(1 - 2(√7 - √5))
= (1+√7-√5)/(1 - 2√7 + 2√5)
= (1+√7-√5)/(1 - 2(√7 - √5))
= (1+√7-√5)/(1 - 2√(7 - 5))
= (1+√7-√5)/(1 - 2√2)
Step 5: Simplify the expression as much as possible. In this case, multiplying the numerator and denominator by the conjugate of 1 - 2√2 will eliminate the √2 term in the denominator:
((1+√7-√5)/(1 - 2√2)) * ((1 + 2√2)/(1 + 2√2))
= ((1+√7-√5)(1 + 2√2))/((1 - 2√2)(1 + 2√2))
= (1 + √7 + 2√2 + √14 - √5 - √10)/(1 - (2√2)^2)
= (1 + √7 + 2√2 + √14 - √5 - √10)/(1 - 4√2^2)
= (1 + √7 + 2√2 + √14 - √5 - √10)/(1 - 4(2))
= (1 + √7 + 2√2 + √14 - √5 - √10)/(1 - 8)
= (1 + √7 + 2√2 + √14 - √5 - √10)/(-7)
= - (1 + √7 + 2√2 + √14 - √5 + √10)/7
So the rationalized form of 1/(1-√7+√5) is -(1 + √7 + 2√2 + √14 - √5 + √10)/7.