Rationales the Denominator and Simplify: (4sqrt3 + 5sqrt2)/(Sqrt48 + Sqrt18) - Mathematics

Rationales the denominator and simplify:

(4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18)

Solution

We know that rationalization factor for sqrt48 + sqrt18 is sqrt48 - sqrt18. We will multiply numerator and denominator of the given expression (4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18) by sqrt48 - sqrt18 to get

(4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18) xx (sqrt48 - sqrt18)/(sqrt48 - sqrt18) = (4 xx sqrt3 xx sqrt48 - 4 sqrt3 xx sqrt18 + 5 xx sqrt2 xx sqrt48 - 5 xx sqrt2 xx sqrt18)/((sqrt48)^2 - (sqrt18)^2)

 = (4sqrt(3 xx 48) - 4 xx sqrt(3 xx 18) + 5 xx sqrt(2 xx 48) - 5 xx sqrt(2 xx 18))/(48  - 18)

= (4sqrt144 - 4sqrt54 + 5sqrt(96) - 5sqrt36)/30

= (4 xx 12 - 4 xx sqrt9 xx sqrt6 + 5 xx sqrt16 xx sqrt6 - 5sqrt36)/30

= (48 - 4 xx 3 xx sqrt6 + 5 xx 4 xx sqrt6 - 5 xx 6)/30

= (48 - 12sqrt6 + 20sqrt6 - 30)/30

= (18 + 8sqrt6)/30

= (9 + 4sqrt6)/15

Hence the given expression is simplified to (9 + 4sqrt6)/15

Concept: Operations on Real Numbers
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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 3 Rationalisation
Exercise 3.2 | Q 4.5 | Page 14

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