Simplify. See Example 9. - √2 3——— √7 3

Verified step by step guidance

1. Step 1: Recognize that the expression is a fraction with a square root in the denominator: \( \frac{\frac{\sqrt{2}}{3}}{\frac{\sqrt{7}}{3}} \).
2. Step 2: Simplify the fraction by multiplying both the numerator and the denominator by 3 to eliminate the fractions: \( \frac{\sqrt{2}}{\sqrt{7}} \).
3. Step 3: Rationalize the denominator by multiplying both the numerator and the denominator by \( \sqrt{7} \): \( \frac{\sqrt{2} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} \).
4. Step 4: Simplify the expression: \( \frac{\sqrt{14}}{7} \).
5. Step 5: The expression is now simplified with a rational denominator.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, to rationalize a denominator like √7, you would multiply by √7/√7.

Simplifying Radicals

Simplifying radicals means reducing a square root to its simplest form, which involves factoring out perfect squares. For instance, √8 can be simplified to 2√2 because 8 = 4 × 2, and √4 is a perfect square. This process helps in making calculations easier and clearer.

Fraction Operations

Understanding fraction operations is crucial for simplifying expressions involving fractions. This includes knowing how to add, subtract, multiply, and divide fractions, as well as how to combine fractions with different denominators. In the context of the given problem, it is important to manage the fractions correctly to achieve a simplified result.