What is the simplified value of \frac{6}{\sqrt{7} - 1} - \frac{6}{\sqrt{7} + 1}? — JAMB UTME

What is the simplified value of $\frac{6}{\sqrt{7} - 1} - \frac{6}{\sqrt{7} + 1}$ ?

Combining the fractions over the common denominator

(\sqrt{7} - 1)(\sqrt{7} + 1) = 6

yields a numerator of

6(\sqrt{7} + 1) - 6(\sqrt{7} - 1) = 12

. Dividing 12 by 6 gives 2.

Find a common denominator for the two fractions

The common denominator is

(\sqrt{7} - 1)(\sqrt{7} + 1) = (\sqrt{7})^2 - 1^2 = 7 - 1 = 6

Multiplying conjugate surds eliminates the radical in the denominator.

Combine the numerators over the common denominator

\frac{6(\sqrt{7} + 1) - 6(\sqrt{7} - 1)}{6}

Adjust each numerator by multiplying by the conjugate of its denominator.

Expand and simplify the numerator

6\sqrt{7} + 6 - 6\sqrt{7} + 6 = 12

The

6\sqrt{7}

terms cancel out:

6\sqrt{7} - 6\sqrt{7} = 0

, leaving

6 - (-6) = 12

Divide the simplified numerator by the denominator

\frac{12}{6} = 2

Simplify the final fraction to obtain an integer value.

Rationalisation and Subtraction of Surd Expressions

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