Rationalize the denominator using the conjugate.

1/1-√7+√5

please show all the steps and explain it

1/(1-√7+√5)

multiply top and bottom by 1-√7-√5

(1-√7-√5)/((1-√7)^2 - 5)
= (1-√7-√5)/(1-2√7+7-5)
= (1-√7-√5)/(3-2√7)
now multiply by 3+2√7
(1-√7-√5)(3+2√7)/(9-28)
(1-√7-√5)(3+2√7)/-19

Now just expand and simply the top to get

(11+3√5+√7+2√35)/19

Not sure I don't prefer the original expression

thankyouuu !!

To rationalize the denominator using the conjugate, we need to multiply the numerator and denominator of the fraction by the conjugate of the denominator.

The conjugate of a binomial with a radical involves changing the sign between the terms. In this case, the conjugate of 1-√7+√5 would be 1+√7-√5.

Now, let's perform the multiplication.

First, multiply the numerator and denominator by the conjugate:

(1/1-√7+√5) * (1+√7-√5)/(1+√7-√5)

Next, let's simplify the denominator by applying the distributive property:

(1*(1+√7-√5) + √7*(1+√7-√5) - √5*(1+√7-√5))/(1+√7-√5)

We can simplify the numerator further by combining like terms inside parentheses:

(1 + √7 + √7 + 7 - √5 - √5 - √35 + 5 - √35)/(1+√7-√5)

Now, simplify the numerator:

(13 + 2√7 - 2√5 - 2√35)/ (1+√7-√5)

And we have successfully rationalized the denominator using the conjugate!