If the expression \frac{\sqrt{48}}{\sqrt{3} + \sqrt{2}} + \frac{\sqrt{72}}{\sqrt{3} - \sqr… — JAMB UTME

If the expression $\frac{\sqrt{48}}{\sqrt{3} + \sqrt{2}} + \frac{\sqrt{72}}{\sqrt{3} - \sqrt{2}}$ is simplified into the form $m + n\sqrt{6}$ , where $m$ and $n$ are integers, find the value of $m + n$ .

Answer: 26

Answer

The simplified expression is

24 + 2\sqrt{6}

, giving

m = 24

and

n = 2

, so

m + n = 26

Simplifying

\sqrt{48}

4\sqrt{3}

and

\sqrt{72}

6\sqrt{2}

allows rationalization of each fraction by its conjugate. The first fraction becomes

12 - 4\sqrt{6}

and the second becomes

12 + 6\sqrt{6}

. Adding these expressions results in

24 + 2\sqrt{6}

, so

m = 24

and

n = 2

, giving

m + n = 26

Step-by-Step Solution

Simplify the radical numerators

\sqrt{48} = 4\sqrt{3}

and

\sqrt{72} = 6\sqrt{2}

Decomposing surds into perfect square factors simplifies subsequent algebraic expansion.

Rationalize the first term

\frac{4\sqrt{3}}{\sqrt{3} + \sqrt{2}}

12 - 4\sqrt{6}

Multiplying the numerator and denominator by the conjugate

(\sqrt{3} - \sqrt{2})

removes the surd from the denominator using the difference of squares

(\sqrt{3})^2 - (\sqrt{2})^2 = 1

Rationalize the second term

\frac{6\sqrt{2}}{\sqrt{3} - \sqrt{2}}

12 + 6\sqrt{6}

Multiplying the numerator and denominator by the conjugate

(\sqrt{3} + \sqrt{2})

yields a rational denominator of

1

Combine like surd terms

24 + 2\sqrt{6}

Summing the rational components

(12 + 12 = 24)

and combining similar surd terms

(-4\sqrt{6} + 6\sqrt{6} = 2\sqrt{6})

Calculate the target sum

m + n

26

Matching coefficients with

m + n\sqrt{6}

gives

m = 24

and

n = 2

, yielding

24 + 2 = 26

Key Concept

Rationalization of Binomial Denominators using Conjugates

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