Problem 48 - College Algebra

Question

Rationalize the denominator. $$\frac{\sqrt{7}}{\sqrt{3}}$$

Step-by-Step Solution

Answer

\(\frac{\sqrt{21}}{3}\)

Step 1: Identify the radicands

First, each number under the root is called a 'radicand'. So in the fraction \(\frac{\sqrt{7}}{\sqrt{3}}\), the radicands are \(7\) and \(3\).

Step 2: Multiply by Conjugate

When rationalizing the denominator, multiply the numerator and denominator by the square root of the denominator's radicand. This would look like \(\frac{\sqrt{7}}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}\). It is allowed to multiply like this since it is just equivalent to multiplying by 1, which does not change the fraction.

Step 3: Simplify the Fraction

Simplify the fraction by applying the multiplication. Multiplying two square roots results to the radicand, hence, \(\sqrt{7}\times\sqrt{3}=\sqrt{21}\) and \(\sqrt{3}\times\sqrt{3}=\sqrt{9}=3\). This provides us the fraction \(\frac{\sqrt{21}}{3}\).