Rationalize the denominator: 5 /4-squareroot 5

(five over four minus square root five)

Please explain

since (a-b)(a+b) = a^2 - b^2,

(4-sqrt(5))(4+sqrt(5) = 16-5 = 11

so, multiply top and bottom by 4+sqrt(5) to get

5(4+sqrt(5))/11 = (20 + 5sqrt(5))/11

Thank you so much!! It really helped me understand!

To rationalize the denominator, we need to get rid of the square root in the denominator. Here's how you can do it step by step:

Step 1: Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate is formed by changing the sign of the term with the square root. In this case, the conjugate of the denominator, 4 - √5, is 4 + √5.

Step 2: Apply the distributive property and multiply the numerator and denominator.

5 * (4 + √5) / (4 - √5) * (4 + √5)

Step 3: Simplify the numerator by using the distributive property.

20 + 5√5 / (4 - √5) * (4 + √5)

Step 4: Simplify the denominator by multiplying the binomials using the FOIL method.

(4 - √5) * (4 + √5) = 16 - 5

The denominator simplifies to 16 - 5 = 11, so the expression becomes:

20 + 5√5 / 11

Therefore, the rationalized form of the given expression 5 / (4 - √5) is (20 + 5√5) / 11.