Problem 46 - College Algebra

Question

Rationalize the denominator. $$\frac{2}{\sqrt{10}}$$

Step-by-Step Solution

Answer

The rationalized form of \( \frac{2}{\sqrt{10}} \) is \( \frac{\sqrt{10}}{5} \).

Step 1: Analyze the Fraction

The given fraction is \( \frac{2}{\sqrt{10}} \). The aim is to remove the square root from the denominator. The method used to do this is to multiply the fraction by a clever form of one, that doesn't change the value of the fraction, but helps to achieve our goal.

Step 2: Form the Clever 1

The clever form of one in this case would be \( \frac{\sqrt{10}}{\sqrt{10}} \), since multiplying any number by its square root gives a rational number without a square root.

Step 3: Multiply

Multiply the fraction \( \frac{2}{\sqrt{10}} \) by the clever form of 1, \( \frac{\sqrt{10}}{\sqrt{10}} \). We get \( \frac{2*\sqrt{10}}{\sqrt{10}*\sqrt{10}} \).

Step 4: Simplify the Fraction

Simplify the fraction \( \frac{2*\sqrt{10}}{\sqrt{10}*\sqrt{10}} \) to get \( \frac{2*\sqrt{10}}{10} \).

Step 5: Further Simplification

The fraction \( \frac{2*\sqrt{10}}{10} \) can be further simplified to \( \frac{2\sqrt{10}}{10} \). This is because 2 and 10 have a common factor of 2 that can be divided throughout. So, the final simplified fraction after rationalizing the denominator is \( \frac{\sqrt{10}}{5} \).