If \frac{4\sqrt{3}}{\sqrt{7} + \sqrt{3}} + \frac{4\sqrt{7}}{\sqrt{7} - \sqrt{3}} = p + q\s… — JAMB UTME

If $\frac{4\sqrt{3}}{\sqrt{7} + \sqrt{3}} + \frac{4\sqrt{7}}{\sqrt{7} - \sqrt{3}} = p + q\sqrt{21}$ , where $p$ and $q$ are integers, what is the value of $p + q$ ?

Answer: 6

Answer

The value of

p + q

is 6.

Rationalising each fraction yields

(\sqrt{21} - 3)

and

(7 + \sqrt{21})

. Summing these expressions gives

4 + 2\sqrt{21}

. Comparing this to

p + q\sqrt{21}

yields

p = 4

and

q = 2

, so

p + q = 6

Step-by-Step Solution

Rationalise the first term

\frac{4\sqrt{3}}{\sqrt{7} + \sqrt{3}}

by multiplying the numerator and denominator by the conjugate

(\sqrt{7} - \sqrt{3})

\frac{4\sqrt{3}(\sqrt{7} - \sqrt{3})}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{4\sqrt{21} - 12}{7 - 3} = \frac{4\sqrt{21} - 12}{4} = \sqrt{21} - 3

Multiplying by the conjugate eliminates the surd from the denominator using the difference of two squares.

Rationalise the second term

\frac{4\sqrt{7}}{\sqrt{7} - \sqrt{3}}

by multiplying the numerator and denominator by the conjugate

(\sqrt{7} + \sqrt{3})

\frac{4\sqrt{7}(\sqrt{7} + \sqrt{3})}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{28 + 4\sqrt{21}}{7 - 3} = \frac{28 + 4\sqrt{21}}{4} = 7 + \sqrt{21}

Conjugate rationalisation simplifies the second fraction into linear surd terms.

Add the two simplified expressions together and equate to

p + q\sqrt{21}

(\sqrt{21} - 3) + (7 + \sqrt{21}) = 4 + 2\sqrt{21}

Combining like surd terms yields the simplified form

p + q\sqrt{21}

Identify the values of

p

and

q

and evaluate

p + q

p = 4, q = 2 \implies p + q = 4 + 2 = 6

Equating the rational parts gives

p = 4

and the irrational coefficients gives

q = 2

Key Concept

Rationalisation of surds with binomial denominators

Alternative Method

Combine the two fractions directly over the common denominator

(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = 4

: \frac{4\sqrt{3}(\sqrt{7} - \sqrt{3}) + 4\sqrt{7}(\sqrt{7} + \sqrt{3})}{4} = \frac{4\sqrt{21} - 12 + 28 + 4\sqrt{21}}{4} = \frac{16 + 8\sqrt{21}}{4} = 4 + 2\sqrt{21}.

Estimated Time:1m 30s

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