What is the simplified numerical value of \frac{\sqrt{12}}{\sqrt{3} + 1} + \frac{\sqrt{12}… — JAMB UTME

What is the simplified numerical value of $\frac{\sqrt{12}}{\sqrt{3} + 1} + \frac{\sqrt{12}}{\sqrt{3} - 1}$ ?

Answer: 6

The simplified numerical value of the given expression is 6.

Combining the fractions using their common conjugate denominator

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

leads to a numerator of

2\sqrt{3}[(\sqrt{3}-1)+(\sqrt{3}+1)] = 2\sqrt{3}(2\sqrt{3}) = 12

. Dividing 12 by 2 yields the final answer of 6.

Simplify the surd in the numerator

\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

Simplifying surds into basic form makes subsequent calculations simpler.

Combine the fractions by finding a common denominator

\frac{2\sqrt{3}(\sqrt{3}-1) + 2\sqrt{3}(\sqrt{3}+1)}{(\sqrt{3}+1)(\sqrt{3}-1)}

Multiplying denominators forms a conjugate pair, which rationalizes the combined denominator.

Simplify the numerator and the denominator

\text{Numerator} = 2\sqrt{3}(\sqrt{3}-1 + \sqrt{3}+1) = 2\sqrt{3}(2\sqrt{3}) = 12; \quad \text{Denominator} = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2

Expanding the numerator combines like surd terms, and using the difference of squares simplifies the denominator to a rational integer.

Perform the final division

\frac{12}{2} = 6

Dividing the simplified numerator by the rationalized denominator gives the final integer value.

Rationalization of binomial denominators using conjugate surds

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