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

Verified step by step guidance

1. Rewrite the expression clearly as a fraction: \(\frac{-\sqrt{2} \times 3}{\sqrt{7} \times 3}\).
2. Notice that the factor 3 appears in both numerator and denominator, so you can simplify by canceling out the 3: \(\frac{-\sqrt{2}}{\sqrt{7}}\).
3. To simplify the fraction with square roots in numerator and denominator, multiply numerator and denominator by \(\sqrt{7}\) to rationalize the denominator: \(\frac{-\sqrt{2}}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}\).
4. Use the property \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\) to rewrite numerator and denominator: numerator becomes \(-\sqrt{2 \times 7}\) and denominator becomes \(\sqrt{7 \times 7}\).
5. Simplify the denominator \(\sqrt{7 \times 7} = 7\), so the expression becomes \(\frac{-\sqrt{14}}{7}\). This is the simplified form.

Key Concepts

Simplifying Radicals

Simplifying radicals involves reducing the expression under the square root to its simplest form by factoring out perfect squares. This process makes it easier to perform arithmetic operations and compare radical expressions.

Rationalizing the Denominator

Rationalizing the denominator means eliminating any radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable radical expression. This results in a simplified expression with a rational denominator.

Properties of Square Roots

Square root properties, such as √a * √b = √(a*b) and √(a/b) = √a / √b, allow manipulation and simplification of expressions involving radicals. Understanding these properties is essential for correctly simplifying and rationalizing expressions.