Problem 48 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{2 a^{3} b}}{\sqrt{14 a}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{\sqrt{2 a^{3} b}}{\sqrt{14 a}}$$. Answer: The simplified expression is $$\frac{a\sqrt{ab}}{\sqrt{7a}}$$.

Step 1: Break up the square roots

Rewrite the given expression as a product of square roots. $$ \frac{\sqrt{2 a^{3} b}}{\sqrt{14 a}} = \frac{\sqrt{2}\cdot\sqrt{a^3}\cdot\sqrt{b}}{\sqrt{14}\cdot\sqrt{a}} $$

Step 2: Simplify the terms inside the square roots

Factor out the perfect squares inside the square roots and simplify the expression. $$ \frac{\sqrt{2}\cdot\sqrt{a^3}\cdot\sqrt{b}}{\sqrt{14}\cdot\sqrt{a}} = \frac{\sqrt{2}\cdot a\sqrt{a}\cdot \sqrt{b}}{\sqrt{2}\cdot \sqrt{7}\cdot \sqrt{a}} $$

Step 3: Cancel out common factors

Cancel out the common factors of the numerator and the denominator. $$ \frac{\sqrt{2}\cdot a\sqrt{a}\cdot \sqrt{b}}{\sqrt{2}\cdot \sqrt{7}\cdot \sqrt{a}} = \frac{a\sqrt{a}\cdot \sqrt{b}}{\sqrt{7}\cdot \sqrt{a}} $$

Step 4: Combine square roots

Combine the square roots in the numerator and the denominator. $$ \frac{a\sqrt{a}\cdot \sqrt{b}}{\sqrt{7}\cdot \sqrt{a}} = \frac{a\sqrt{ab}}{\sqrt{7a}} $$

Step 5: Final simplified expression

Write the final simplified expression. $$ \frac{\sqrt{2 a^{3} b}}{\sqrt{14 a}} = \frac{a\sqrt{ab}}{\sqrt{7a}} $$

Key Concepts

Square Roots

Understanding square roots is crucial in simplifying radical expressions. A square root, denoted as \( \sqrt{} \), refers to a value that, when multiplied by itself, will give the original number. For example, the square root of 9 is 3, because \( 3 \times 3 = 9 \). In terms of algebraic expressions, finding the square root involves identifying parts of the expression that can be paired and simplified.

• Perfect Square Factors: These are numbers or variables within an expression that have a whole number as their square root. For instance, \( a^2 \) is a perfect square because \( \sqrt{a^2} = a \).
• Simplification Process: When you encounter \( \sqrt{a^3} \), it can be rewritten as \( a\sqrt{a} \), by extracting \( a^2 \) and leaving \( a \) under the root.

Breaking down the square roots into simpler components helps in identifying and removing common factors later in the process.

Factorization

Factorization is the process of breaking down numbers or expressions into multiples that combine to give the original value. In the context of simplifying radicals, factorization helps us identify components that are perfect squares or that have common bases which can be further simplified.

• Finding Factors: To factorize \( 14 \), recognize it as \( 2 \times 7 \). This helps in breaking down \( \sqrt{14} \) into \( \sqrt{2} \cdot \sqrt{7} \), making simplification easier.
• Algebraic Factorization: For \( a^3 \), factorizing gives \( a^2 \cdot a \). This allows us to take \( a \) out of the square root, simplifying \( \sqrt{a^3} \) to \( a\sqrt{a} \).

By recognizing these factors within an expression, you can start simplifying each component individually, making the whole expression simpler to handle.

Canceling Common Factors

Canceling common factors is a vital step in simplifying fractions of radical expressions. When both the numerator and the denominator share a common factor, it can be removed, reducing the expression to its simplest form.

• Identifying Common Factors: Look for similar terms in both the top and bottom of the fraction. In the expression \( \frac{\sqrt{2} \cdot a \sqrt{a} \cdot \sqrt{b}}{\sqrt{2} \cdot \sqrt{7} \cdot \sqrt{a}} \), \( \sqrt{2} \) and \( \sqrt{a} \) are present in both the numerator and denominator.
• Canceling Process: By canceling these terms, you simplify the fraction, reducing the complexity of the entire expression.

This step clears away unnecessary parts of the expression, focusing only on the essential elements that need further simplification.

Radical Expressions

Radical expressions involve roots of numbers or variables, often appearing complex at first glance. Understanding how to work with these expressions is key to simplifying them.

• Combining Radicals: You can multiply and divide radical expressions by combining their contents. For example, \( \sqrt{a} \cdot \sqrt{b} \) can be combined into \( \sqrt{ab} \).
• Final Simplifications: After combining and canceling common factors, the expression might simplify further, as seen with \( \frac{a\sqrt{ab}}{\sqrt{7a}} \), where attempts to simplify often result in a cleaner end expression.

These skills help unravel the complexities within radical expressions, allowing for more straightforward computations and results.