Simplify. See Example 9. √7 5 ——— √3 10

Verified step by step guidance

1. Step 1: Recognize that the expression is a fraction with a square root in the numerator and denominator: $\frac{\sqrt{7}\cdot 5}{\sqrt{3}\cdot 10}$.
2. Step 2: Simplify the fraction by multiplying the numerators and the denominators: $\frac{5\sqrt{7}}{10\sqrt{3}}$.
3. Step 3: Simplify the fraction by dividing both the numerator and the denominator by their greatest common factor, which is 5: $\frac{\sqrt{7}}{2\sqrt{3}}$.
4. Step 4: Rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{3}$: $\frac{\sqrt{7}\cdot \sqrt{3}}{2\sqrt{3}\cdot \sqrt{3}}$.
5. Step 5: Simplify the expression by using the property $\sqrt{a}\cdot \sqrt{a}=a$ to get $\frac{\sqrt{21}}{6}$.

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 done by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, if the denominator is √3, multiplying by √3/√3 will help achieve this.

Simplifying Radicals

Simplifying radicals means reducing a square root to its simplest form. This involves factoring out perfect squares from under the radical sign. For instance, √12 can be simplified to 2√3, as 12 = 4 × 3, and 4 is a perfect square. This concept is essential for making calculations easier and clearer.

Fraction Operations

Understanding how to perform operations with fractions is crucial for simplifying expressions. This includes adding, subtracting, multiplying, and dividing fractions, as well as finding a common denominator when necessary. In the given expression, combining the fractions correctly after simplification is key to arriving at the final answer.