Problem 47 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{\frac{2}{3}}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{\sqrt{6}}{3}\).

Step 1: Rationalize the Denominator

To simplify \(\sqrt{\frac{2}{3}}\), first eliminate the square root from the denominator by multiplying both the numerator and the denominator by \(\sqrt{3}\). This gives: \(\frac{\sqrt{2} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}\).

Step 2: Multiply the Radicals

Multiply the radicals in the numerator and the denominator. This results in: \(\frac{\sqrt{6}}{3}\). The denominator becomes 3 because \(\sqrt{3} \times \sqrt{3} = 3\).

Step 3: Simplify the Expression

The expression \(\frac{\sqrt{6}}{3}\) is in its simplest form as there are no common factors to further simplify.

Key Concepts

Rationalizing the Denominator

When you encounter a fraction with a radical in the denominator, it's often necessary to "rationalize" it. Rationalizing the denominator involves transforming the fraction so that there are no radicals in the denominator.
To do this, multiply both the numerator and the denominator by the same radical term present in the denominator.

• For example, in the expression \(\sqrt{\frac{2}{3}}\), the denominator is \(\sqrt{3}\).
• Multiply both the numerator and the denominator by \(\sqrt{3}\) to eliminate the radical in the denominator.

This results in a new expression:
\(\frac{\sqrt{2} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}\).
This process effectively grants us a denominator that is a simple integer rather than a square root, completing the rationalization step.

Multiplying Radicals

Once the denominator is rationalized, it's time to multiply the radicals in both the numerator and the denominator. Radicals can be multiplied together quite similarly to regular numbers. Here's how the process works:- When multiplying square roots, multiply the numbers inside the root.
- Use the property: \(\sqrt{a} \times \sqrt{b} = \sqrt{a\times b}\).
In our example:

• The numerator \(\sqrt{2} \times \sqrt{3}\) results in \(\sqrt{6}\).
• For the denominator, \(\sqrt{3} \times \sqrt{3}\) turns into 3 because \(\sqrt{3\times 3} = \sqrt{9} = 3\).

This results in the fraction \(\frac{\sqrt{6}}{3}\). That's how multiplying radicals makes the expression more simplified and manageable.

Simplifying Expressions

Now that we've multiplied the radicals and rationalized the denominator, the final task is simplifying the expression.Simplification means reducing the expression to its simplest form, ensuring there's nothing more that can be factored or divided out from either the numerator or the denominator.
Consider our ending expression: \(\frac{\sqrt{6}}{3}\).- Check if there are common factors in the numerator and denominator. - Here, \(\sqrt{6}\) cannot be further simplified because \(6\) itself is a product of \(2 \times 3\), so there are no perfect square factors to extract.- The denominator is a simple integer (3), so further simplification is not possible.Reaching this stage, we conclude the expression \(\frac{\sqrt{6}}{3}\) is already in its simplest form.
Simplifying expressions ensures complete clarity and conciseness in the final result.