Rationalise the denominator of the following : (4√3 + 5√2) / (√48 + √18)

Solution:

Given, the expression is (4√3 + 5√2) / (√48 + √18)

We have to rationalise the denominator

√48 = √16 × 3 = 4√3

√18 = √9 × 2 = 3√2

(4√3 + 5√2) / (√48 + √18) = (4√3 + 5√2) / (4√3 + 3√2)

To rationalise we have to take conjugate,

(4√3 + 5√2) / (4√3 + 3√2) = (4√3 + 5√2) / (4√3 + 3√2) × (4√3 - 3√2) / (4√3 - 3√2)

= (4√3 + 5√2)(4√3 - 3√2) / (4√3 + 3√2)(4√3 - 3√2)

By using algebraic identity,

(a² - b²) = (a - b)(a + b)

(4√3 + 3√2)(4√3 - 3√2) = (4√3)² - (3√2)²

= 16(3) - 9(2)

= 48 - 18

= 30

So, (4√3 + 5√2)(4√3 - 3√2) / (4√3 + 3√2)(4√3 - 3√2) = (4√3 + 5√2)(4√3 - 3√2) / 30

By multiplicative and distributive property,

(4√3 + 5√2)(4√3 - 3√2) = 4√3(4√3) - 4√3(3√2) + 5√2(4√3) - 5√2(3√2)

= 16(3) - 12√6 + 20√6 - 15(2)

= 48 - 30 - 12√6 + 20√6

= 18 + 8√6

Now, (4√3 + 5√2)(4√3 - 3√2) / 30 = (18 + 8√6) / 30

= 2(9 + 4√6) / 30

= 9 + 4√6 / 15

Therefore, (4√3 + 5√2) / (√48 + √18) = (9 + 4√6) / 15

✦ Try This: Rationalise the denominator of the following : (3√2 + 5√3) / (√24 + √27)

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1

NCERT Exemplar Class 9 Maths Exercise 1.3 Problem 10(ix)

Rationalise the denominator of the following : (4√3 + 5√2) / (√48 + √18)

Summary:

Rationalizing the denominator means the process of moving a root, for instance, a cube root or a square root from the bottom of a fraction (denominator) to the top of the fraction (numerator). On rationalising the denominator of (4√3 + 5√2) / (√48 + √18) we get (9 + 4√6) / 15

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