If \sqrt{75} + \sqrt{12} - \sqrt{27} = k\sqrt{3}, what is the value of k? — JAMB UTME

If $\sqrt{75} + \sqrt{12} - \sqrt{27} = k\sqrt{3}$ , what is the value of $k$ ?

Answer: 4

The value of

k

is 4.

Simplifying each surd into its basic form yields

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

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

, and

\sqrt{27} = 3\sqrt{3}

. Combining the coefficients gives

(5 + 2 - 3)\sqrt{3} = 4\sqrt{3}

. Equating

4\sqrt{3}

k\sqrt{3}

shows that

k = 4

Simplify each individual surd by factoring out perfect squares

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

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

\sqrt{27} = 3\sqrt{3}

To combine surds through addition or subtraction, they must be converted to similar surds.

Combine the coefficients of the like surds

5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} = (5 + 2 - 3)\sqrt{3} = 4\sqrt{3}

Like terms with the same radical factor

\sqrt{3}

can be added and subtracted directly.

Compare the resulting coefficient with

k\sqrt{3}

k = 4

By direct comparison of coefficients of

\sqrt{3}

k

equals 4.

Simplification and combining of similar surds

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