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

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Identify the expression to simplify: $\sqrt{\frac{1}{49}}$.

Recall that the square root of a quotient can be written as the quotient of the square roots: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Apply this property to rewrite the expression as $\frac{\sqrt{1}}{\sqrt{49}}$.

Simplify the square roots individually: $\sqrt{1} = 1$ and $\sqrt{49} = 7$.

Write the simplified expression as $\frac{1}{7}$.

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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.

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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.

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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.

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Imaginary Roots with the Square Root Property

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Domanda del libro di testo

Multiply or divide as indicated. $\frac{x + 5}{7}\) $\div$ \(\frac{4x + 20}{9}$