Simplify the expression \frac{\sqrt{3}}{\sqrt{3} - 1} - \frac{\sqrt{3}}{\sqrt{3} + 1}. — JAMB UTME

Simplify the expression $\frac{\sqrt{3}}{\sqrt{3} - 1} - \frac{\sqrt{3}}{\sqrt{3} + 1}$ .

\sqrt{3}

Combining the fractions using the common denominator

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

gives a numerator of

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

. Dividing

2\sqrt{3}

2

yields the simplified answer

\sqrt{3}

Find a common denominator for the two fractions

The common denominator is

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

The denominators are conjugate surds, so their product simplifies to a rational number using the difference of two squares.

Express the numerator over the common denominator

Numerator

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

Multiply each numerator by the missing factor of the common denominator.

Expand and simplify the numerator

Numerator

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

Distribute

\sqrt{3}

and handle the subtraction sign carefully.

Divide the simplified numerator by the common denominator

\frac{2\sqrt{3}}{2} = \sqrt{3}

Cancel the common factor of 2.

Rationalisation of denominators and algebraic manipulation of surd fractions

Tahmini Süre:1m 15s

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