الرياضيات (بالإنجليزية) الصف الثاني الإعدادي، الأسبوع 3: Additive and multiplicative inverses, conjugates and rationalising a denominator

Radicals

الأسبوع 3 · العمليات على الجذور: الحدود الجذرية المتشابهة والمعكوسات والمرافق وقوانين الجذور التربيعية والتكعيبية · 3 دروس

Additive and multiplicative inverses, conjugates and rationalising a denominator

Mathematics Prep 2 Term 1, Unit 1, Lessons 3–4

GoalFind the additive and multiplicative inverse of a radical number, and use conjugates to remove a root from a denominator.

New words

additive inverse · the number that gives 0 when added: the inverse of a is −amultiplicative inverse · the number that gives 1 when multiplied: the inverse of a is 1/a (a ≠ 0)conjugates · a + √b and a − √brationalise · remove the root from a denominator

Explanation

The additive inverse of a real number x is −x, so the inverse of 3√2 is −3√2 and the inverse of √5 − 1 is 1 − √5. The multiplicative inverse is 1/x for x ≠ 0. For 3√2 it is 1/(3√2); multiply top and bottom by √2 to remove the root from the denominator: √2/(3 × 2) = √2/6. For 2 − √3 we use its conjugate 2 + √3. A pair a + √b and a − √b multiplies to a whole number, a² − b: (2 − √3)(2 + √3) = 4 − 3 = 1. So 1/(2 − √3) = (2 + √3)/((2 − √3)(2 + √3)) = 2 + √3. Multiplying top and bottom by the same number does not change the value of a fraction.

ƒFormula

(a + √b)(a − √b) = a² − b

The product of conjugates has no root.

Worked examples

Find the multiplicative inverse of 3√2 and of √5 + 1. First: 1/(3√2) = √2/(3√2 × √2) = √2/6. Second: 1/(√5 + 1) = (√5 − 1)/((√5 + 1)(√5 − 1)) = (√5 − 1)/(5 − 1) = (√5 − 1)/4.

Let x = √5 − 2 and y = √5 + 2. Then x − y = −4 and x + y = 2√5, so x² − y² = (x − y)(x + y) = −4 × 2√5 = −8√5. Also x² = 9 − 4√5 and y² = 9 + 4√5, so x² + y² = 18.

A rectangle has sides 3 + √2 and 3 − √2 cm. Its area is (3 + √2)(3 − √2) = 9 − 2 = 7 cm², and its perimeter is 2(3 + √2 + 3 − √2) = 12 cm. The roots cancel in both.

Class activity

Calculator test. Find the decimals of 1/(2 − √3) and 2 + √3. They should agree: 3.7320…. Then try 1/(√5 + 1) and (√5 − 1)/4: both should show 0.3090….

Practice

Write the additive inverse of 4 − √7.

Find the multiplicative inverse of 5√3.

Rationalise 1/(3 + √5).

Simplify 3/(√7 − 2).

Let x = √3 + 1 and y = √3 − 1. Find xy and x² + y².