Use the product and quotient rules for radicals to rewrite each expression. See Example 4. 30√10 / 5√2

Verified step by step guidance

1. Identify the given expression as a quotient involving radicals: \(\frac{30\sqrt{10}}{5\sqrt{2}}\).
2. Apply the quotient rule for radicals, which states that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), to rewrite the expression as \(\frac{30}{5} \times \sqrt{\frac{10}{2}}\).
3. Simplify the numerical fraction \(\frac{30}{5}\) to get 6, so the expression becomes \(6 \times \sqrt{\frac{10}{2}}\).
4. Simplify the fraction inside the radical \(\frac{10}{2}\) to get 5, so the expression is now \(6 \times \sqrt{5}\).
5. Write the final simplified expression as \(6\sqrt{5}\), which uses the product and quotient rules for radicals.

Key Concepts

Product Rule for Radicals

The product rule states that the square root of a product equals the product of the square roots: √(a * b) = √a * √b. This allows simplification by breaking down radicals into factors, making it easier to combine or simplify expressions involving roots.

Quotient Rule for Radicals

The quotient rule states that the square root of a quotient equals the quotient of the square roots: √(a / b) = √a / √b, where b ≠ 0. This rule helps in rewriting expressions with radicals in the numerator and denominator, facilitating simplification or rationalization.

Simplifying Radical Expressions

Simplifying radicals involves factoring numbers inside the root to extract perfect squares and reduce the expression. This process often uses the product and quotient rules to rewrite and combine radicals into simpler or more standard forms.