If the surd expression \sqrt{75} - \sqrt{12} + \frac{6}{\sqrt{3}} is simplified to the for… — JAMB UTME

If the surd expression $\sqrt{75} - \sqrt{12} + \frac{6}{\sqrt{3}}$ is simplified to the form $k\sqrt{3}$ , what is the value of $k$ ?

Answer: 5

The value of

k

is 5.

Simplifying

\sqrt{75}

yields

5\sqrt{3}

, simplifying

\sqrt{12}

yields

2\sqrt{3}

, and rationalizing

\frac{6}{\sqrt{3}}

gives

2\sqrt{3}

. Adding these together gives

5\sqrt{3} - 2\sqrt{3} + 2\sqrt{3} = 5\sqrt{3}

. Equating

5\sqrt{3}

k\sqrt{3}

yields

k = 5

Simplify the individual square roots.

\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}

and

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

Factoring out perfect square numbers allows surds to be written in basic radical form.

Rationalize the fractional surd term

\frac{6}{\sqrt{3}}

\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}

Multiplying numerator and denominator by

\sqrt{3}

eliminates the radical from the denominator.

Combine like surd terms.

5\sqrt{3} - 2\sqrt{3} + 2\sqrt{3} = 5\sqrt{3}

Like surds share the same radical factor and can be added or subtracted algebraically.

Equate the simplified expression to

k\sqrt{3}

k = 5

Comparing coefficients of

\sqrt{3}

reveals the value of

k

Surd Simplification and Rationalization of Denominators

Estimated Time:1m 30s

Question