Math: radicals: rationalizing the denominator, part II

Tutoring math, you deal with this topic for high school and calculus. The tutor continues with coverage of rationalizing the denominator.

The following is by my understanding.

Way back in my post from Mar 22, 2014, I began about rationalizing the denominator. However, there is a more complicated scenario, as follows:

Example: rationalize the denominator of (3√2)/(1-2√5)

Solution:

In a case like this, one has to multiply both numerator and denominator by the conjugate of the denominator. (See my post from Jul 27, 2026 about the conjugate.)

The conjugate of the denominator is 1+2√5, so we get

[(3√2)(1+2√5)]/[(1-2√5)(1+2√5)]

By FOIL (see see my post from Nov 26, 2012 about the foil method),

(1-2√5)(1+2√5) = 1+2√5-2√5-4(√5)^2 = 1-4*5 = 1-20 = -19

Meanwhile the numerator becomes

3√2(1+2√5) = 3√2+6√(10)

So we arrive at

-(3√2 + 6√(10))/19

Rationalizing the denominator doesn’t necessarily get rid of a mess; it just rewrites the expression with the radicals out of the denominator.

Hand-held calculators that can do this, and show the answer symbolically, have been available for around twenty years.

Source:

Traverse, K.J., Dalton, L.C., Brunner, V.F., Taylor, A.R. (1977). Using Advanced Algebra. Doubleday Canada Limited.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.