Rationalise the Denominator

What you do

You shape the function y = x * (3/(2 + sqrt(5)) - (3*sqrt(5) - k)) and watch the graph answer.

Challenges to clear

• The conjugate gives 3/(2+√5) = 3√5 − k for some rational k. Slide k until the difference is 0 at x = 2.
• Still 0 at x = 5 — so 3/(2+√5) = 3√5 − 6 exactly, no root left below.
• Your turn, new fraction: 4/(3+√5). Multiply top and bottom by the conjugate (3−√5) — the denominator becomes 3² − 5 = 4. The answer has the form p − q√5; slide q to how many √5 survive.
• Now slide p until the difference is 0 at x = 2 — so 4/(3+√5) = 3 − √5 exactly, with no root left underneath.

Check yourself

Why do we multiply by the conjugate (a - sqrt(b)) when the denominator is (a + sqrt(b))?

If you rationalise 1 / (sqrt(3) - 1), what is the denominator after simplification?

• It uses the difference of squares identity to eliminate the square root in the denominator. — correct
• It makes the denominator zero, which simplifies the fraction.
• It is the standard rule for dividing fractions with roots.
• It converts the irrational number into a decimal.
• 2 — correct
• -2
• sqrt(2)
• 1

Think about it

• Multiply 3/(2+√5) by (√5−2)/(√5−2). The denominator becomes (√5)² − 2² = 1. What is the numerator?