Problem 82 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{12-8 \sqrt{7}}{16} $$

Step-by-Step Solution

Answer

\( \frac{3}{4} - \frac{\sqrt{7}}{2} \)

Step 1: Split the Fraction

We can start by separating the components in the numerator within the fraction: \[ \frac{12}{16} - \frac{8\sqrt{7}}{16} \]

Step 2: Simplify Each Fraction

Next, we simplify each fraction individually. For \( \frac{12}{16} \), both numerator and denominator have a common factor of 4:\[ \frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4} \]For \( \frac{8\sqrt{7}}{16} \), \(8\) and \(16\) have a common factor of 8:\[ \frac{8\sqrt{7}}{16} = \frac{8 \sqrt{7} \div 8}{16 \div 8} = \frac{\sqrt{7}}{2} \]

Step 3: Combine the Simplified Terms

Now that we have the simplified expressions, we combine them back: \[ \frac{3}{4} - \frac{\sqrt{7}}{2} \] Since these terms don't have a common fractional denominator, this is the simplified form of the expression.

Key Concepts

Fraction Simplification

Fraction simplification revolves around reducing fractions to their simplest form. When simplifying a fraction, you aim to find the greatest common factor (GCF) of the numerator and denominator. By dividing both by their GCF, you obtain a smaller, equivalent fraction.
In the step-by-step solution given above, fractions were split and simplified individually. This was achieved by identifying common factors. For instance:

• The fraction \( \frac{12}{16} \) was simplified using the factor of 4, reducing it to \( \frac{3}{4} \).
• The expression \( \frac{8\sqrt{7}}{16} \) was simplified by the factor of 8, resulting in \( \frac{\sqrt{7}}{2} \).

Knowing how to find these common factors enables you to simplify both numerical and algebraic fractions confidently.

Radical Expressions

Radical expressions involve roots, like square roots or cube roots. The expression \( \sqrt{7} \) is a radical expression under a square root, which is often encountered during algebraic simplification.
When working with these expressions, there are special considerations. For instance, if simplifying involves a radical term, it's important to maintain its correct form while reducing the fraction. In the step-by-step solution, \( \frac{8\sqrt{7}}{16} \) was simplified by factoring out a common number from both the numerator and denominator without altering the radicand (the term inside the root).
This teaches us that simplifying radicals entails careful handling. Ensure the integrity of the radical expression as you apply common factor simplification. This practice reduces complexity while keeping radicals intact.

Common Factors

Common factors are numbers that precisely divide two or more numbers. Understanding common factors is crucial to simplifying both real numbers and algebraic expressions.
In the given problem, recognizing common factors simplified the fractions efficiently. Take \( 12 \) and \( 16 \): their greatest common factor is 4. Likewise, for \( 8 \) and \( 16 \), the GCF is 8.

• This knowledge helps reduce fractions quickly, ensuring computations remain simple and manageable.
• Identifying these factors involves basic division and multiplication knowledge, making it accessible even at early learning stages.

To enhance efficiency when simplifiying expressions, always look for these factors. They form the foundation of reducing complex expressions efficiently, as demonstrated by the exercise solution provided.