Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. $\sqrt{\frac{1}{49}}$

Verified step by step guidance

1. Identify the expression to simplify: \(\sqrt{\frac{1}{49}}\).
2. Recall that the square root of a quotient can be written as the quotient of the square roots: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\).
3. Apply this property to rewrite the expression as \(\frac{\sqrt{1}}{\sqrt{49}}\).
4. Simplify the square roots individually: \(\sqrt{1} = 1\) and \(\sqrt{49} = 7\).
5. Write the simplified expression as \(\frac{1}{7}\).

Key Concepts

Quotient Rule for Radicals

The quotient rule for radicals states that the square root of a quotient is equal to the quotient of the square roots, i.e., √(a/b) = √a / √b, provided b ≠ 0. This rule allows simplification of expressions involving square roots of fractions.

Simplifying Square Roots

Simplifying square roots involves finding the prime factorization of the number under the root and extracting perfect squares. For example, √49 = 7 because 49 is a perfect square. Simplification makes expressions easier to interpret and use.

Properties of Square Roots with Positive Variables

When variables are positive (x > 0), the square root of x² simplifies directly to x, avoiding absolute value considerations. This assumption simplifies radical expressions and ensures the principal (non-negative) root is taken.