Week 3 Β· Operations on radicals: like radicals, inverses, conjugates and the laws of square and cube roots Β· 3 lessons
2
Additive and multiplicative inverses, conjugates and rationalising a denominator
Mathematics Prep 2 Term 1, Unit 1, Lessons 3β4
GoalFind the additive and multiplicative inverse of a radical number, and use conjugates to remove a root from a denominator.
New words
additive inverse Β· the number that gives 0 when added: the inverse of a is βamultiplicative inverse Β· the number that gives 1 when multiplied: the inverse of a is 1/a (a β 0)conjugates Β· a + βb and a β βbrationalise Β· remove the root from a denominator
Explanation
The additive inverse of a real number x is βx, so the inverse of 3β2 is β3β2 and the inverse of β5 β 1 is 1 β β5. The multiplicative inverse is 1/x for x β 0. For 3β2 it is 1/(3β2); multiply top and bottom by β2 to remove the root from the denominator: β2/(3 Γ 2) = β2/6. For 2 β β3 we use its conjugate 2 + β3. A pair a + βb and a β βb multiplies to a whole number, aΒ² β b: (2 β β3)(2 + β3) = 4 β 3 = 1. So 1/(2 β β3) = (2 + β3)/((2 β β3)(2 + β3)) = 2 + β3. Multiplying top and bottom by the same number does not change the value of a fraction.
Let x = β5 β 2 and y = β5 + 2. Then x β y = β4 and x + y = 2β5, so xΒ² β yΒ² = (x β y)(x + y) = β4 Γ 2β5 = β8β5. Also xΒ² = 9 β 4β5 and yΒ² = 9 + 4β5, so xΒ² + yΒ² = 18.
A rectangle has sides 3 + β2 and 3 β β2 cm. Its area is (3 + β2)(3 β β2) = 9 β 2 = 7 cmΒ², and its perimeter is 2(3 + β2 + 3 β β2) = 12 cm. The roots cancel in both.
Class activity
Calculator test. Find the decimals of 1/(2 β β3) and 2 + β3. They should agree: 3.7320β¦. Then try 1/(β5 + 1) and (β5 β 1)/4: both should show 0.3090β¦.
Practice
1
Write the additive inverse of 4 β β7.
β(4 β β7) = β7 β 4.
2
Find the multiplicative inverse of 5β3.
1/(5β3) = β3/15.
3
Rationalise 1/(3 + β5).
(3 β β5)/(9 β 5) = (3 β β5)/4.
4
Simplify 3/(β7 β 2).
3(β7 + 2)/(7 β 4) = β7 + 2.
5
Let x = β3 + 1 and y = β3 β 1. Find xy and xΒ² + yΒ².
