Problem 62 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt{45}}{\sqrt{9}}\)

Step-by-Step Solution

Answer

\( \sqrt{5} \)

Step 1: Apply the Quotient Rule for Radicals

The quotient rule for radicals states that \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) for positive numbers \(a\) and \(b\). In this case, apply the rule to obtain: \[ \frac{\sqrt{45}}{\sqrt{9}} = \sqrt{\frac{45}{9}} \].

Step 2: Simplify the Fraction Inside the Radical

Now simplify the fraction \( \frac{45}{9} \). Divide 45 by 9 to get \( 5 \). Thus, \( \sqrt{\frac{45}{9}} = \sqrt{5} \).

Step 3: Simplify the Square Root

Evaluate the square root of 9 separately to confirm that the division is correct: \( \sqrt{9} = 3 \). We've already used this operation correctly; thus, \( \sqrt{\frac{45}{9}} = \sqrt{5} \) is confirmed to be simplified properly.

Key Concepts

Simplifying Radicals

When dealing with radicals, especially in an expression with both a numerator and denominator, simplifying the radicals is crucial for obtaining a clean and simplified result. The process often involves using properties of radicals, such as the quotient rule. This rule allows us to divide square roots by combining the radicands under a single square root. Here, it's done by turning the expression \( \frac{\sqrt{45}}{\sqrt{9}} \) into \( \sqrt{\frac{45}{9}} \). This method can help make your work neater and solve expressions more efficiently.

• Combine radicals to minimize complexity.
• Simplify fractions within a radical to reduce the size of numbers.
• Make use of properties, such as factoring or canceling, when possible.

Transforming the problem into a form that's easier to handle is an essential skill in algebraic problem-solving.

Square Roots

Understanding square roots is fundamental when simplifying expressions involving radicals. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, in our exercise, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). It’s important to identify known square roots in order to simplify expressions quickly and accurately.

• Avoid unnecessary steps by calculating known square roots immediately.
• Recognize perfect squares, like 4, 9, 16, to make calculations easier.
• Break down larger numbers to find factors that are perfect squares.

By mastering square roots, you can solve problems faster and reduce chances of error.

Algebra Problem Solving

Solving algebra problems often involves multiple steps and requires a combination of skills. Applying the quotient rule and simplifying radicals are part of this toolbox. It's important to approach each problem systematically to ensure no details are overlooked. In our exercise, each step builds on the last, requiring careful application of rules and simplification techniques.

• Follow the order of operations: always address operations within radicals first.
• Check each step to confirm results are accurate, as demonstrated by verifying \( \sqrt{9} \).
• Simplify at every opportunity to keep expressions manageable.

By using a structured approach to algebra problem-solving, you can increase accuracy and efficiency in reaching solutions.