Simplify. See Example 9. (√7/5)/(√3/10)

Verified step by step guidance

1. Rewrite the expression as a single fraction: \(\frac{\sqrt{7} \times 5}{\sqrt{3} \times 10}\).
2. Simplify the numerical coefficients by dividing 5 by 10, which reduces to \(\frac{1}{2}\), so the expression becomes \(\frac{\sqrt{7}}{\sqrt{3}} \times \frac{1}{2}\).
3. Combine the square roots in the numerator and denominator into a single square root: \(\frac{\sqrt{7}}{\sqrt{3}} = \sqrt{\frac{7}{3}}\).
4. Multiply the simplified square root by \(\frac{1}{2}\) to get \(\frac{1}{2} \times \sqrt{\frac{7}{3}}\).
5. If desired, rationalize the denominator inside the square root by multiplying numerator and denominator inside the root by 3, resulting in \(\frac{1}{2} \times \sqrt{\frac{21}{9}}\), and then simplify further.

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring out perfect squares. This process makes it easier to perform arithmetic operations with radicals and to compare their values.

Rationalizing the Denominator

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

Properties of Square Roots and Fractions

Understanding that the square root of a fraction equals the fraction of the square roots (√(a/b) = √a / √b) helps in manipulating expressions involving radicals and fractions. This property is essential for simplifying complex radical expressions.