Use the quotient rule to simplify the expressions in Exercises 45-46. √(121/4)

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Identify the expression given: \(\sqrt{\frac{121}{4}}\). This is a square root of a fraction.

Recall the property of square roots that allows you to separate the root of a fraction into the root of the numerator over the root of the denominator: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\).

Apply this property to the expression: \(\sqrt{\frac{121}{4}} = \frac{\sqrt{121}}{\sqrt{4}}\).

Simplify the square roots of the numerator and denominator separately: \(\sqrt{121}\) and \(\sqrt{4}\).

Write the simplified expression as a fraction of the simplified roots: \(\frac{\sqrt{121}}{\sqrt{4}}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 roots of fractions by separating numerator and denominator.

Simplifying Square Roots

Simplifying square roots involves finding the prime factors or perfect squares within the radicand to rewrite the root in simplest form. For example, √121 = 11 because 121 is a perfect square, which helps in reducing expressions to their simplest terms.

Properties of Fractions

Understanding fractions is essential when working with radicals in quotient form. Recognizing that a fraction like 121/4 can be expressed as (121)/(4) helps apply the quotient rule correctly and simplifies the expression by dealing with numerator and denominator separately.