Rationalise 1/(2 + √3)

What you do

You shape the function y = x * (1/(2 + sqrt(3)) - (k - sqrt(3))) and watch the graph answer.

Challenges to clear

• The conjugate says 1/(2+√3) = k − √3 for some whole k. Slide k until the difference is 0 at x = 1.
• Still 0 at x = 3 — so 1/(2+√3) = 2 − √3 exactly.
• Same move, harder numbers: 1/(4+√15). The conjugate (4−√15) makes the denominator 4² − 15 = 1. The answer is p − q√15 — slide q to how many √15 are left.
• Now slide p until the difference is 0 at x = 1 — so 1/(4+√15) = 4 − √15, a fraction with no root below the bar.

Check yourself

Why do we multiply by the conjugate?

• Because (a + b)(a - b) = a^2 - b^2, which removes the square root if b is a surd. — correct
• Because it makes the numerator larger and easier to read.
• Because we must always multiply the top and bottom by the same number.
• Because it changes the value of the fraction to a rational number.

Think about it

• Multiply 1/(2+√3) by (2−√3)/(2−√3). The denominator becomes 4 − 3 = 1. What is left on top?