Problem 47 - Algebra 2

Question

ACT/SAT Which expression is equal to \(\frac{\left(2 x^{2}\right)^{3}}{12 x^{4}} ?\) $$ \begin{array}{ll}{\mathbf{A} \frac{x}{2}} & {\mathbf{C} \frac{1}{2 x^{2}}} \\\ {\mathbf{B} \frac{2 x}{3}} & {\mathbf{D} \frac{2 x^{2}}{3}}\end{array} $$

Step-by-Step Solution

Answer

The expression is equal to \(\frac{2x^2}{3}\), matching option D.

Step 1: Expand the Numerator

First, expand \((2x^2)^3\). This will give \(2^3 \times (x^2)^3 = 8x^6\). The expanded form of the numerator is \(8x^6\).

Step 2: Substitute into the Fraction

Substitute the expanded numerator back into the expression: \(\frac{8x^6}{12x^4}\).

Step 3: Simplify the Fraction

Divide both the numerator and the denominator by the common factor of 4: \(\frac{8x^6}{12x^4} = \frac{2x^6}{3x^4}\).

Step 4: Simplify Exponents

Apply the law of exponents \((\frac{x^a}{x^b} = x^{a-b})\) to simplify the fraction: \(\frac{2x^6}{3x^4} = \frac{2x^{6-4}}{3} = \frac{2x^2}{3}\).

Step 5: Choose the Correct Answer

The simplified form of the expression is \(\frac{2x^2}{3}\), which corresponds with option \(\textbf{D}\) in the given choices.

Key Concepts

Simplifying Expressions

When you simplify expressions, you aim to make them more approachable by reducing them to their simplest form. This process often involves eliminating any unnecessary components or complex fractions without changing the value of the expression itself. In the given exercise, simplifying meant rewriting \( \frac{(2x^2)^3}{12x^4} \) in its most reduced form. A good first step is to handle the numerator and denominator separately. Examine each part to find common factors or elements that can be cancelled out. In this example, you begin by expanding \((2x^2)^3)\) to \(8x^6\), a simpler expression. Substituting this into the fraction helps you to identify that \(8\) and \(12\) share a common factor, which is \(4\). Dividing both the numerator and the denominator by \(4\) results in \(\frac{2x^6}{3x^4}\). This new fraction is much simpler due to smaller numbers in the numerator and denominator, paving the path to further simplification using exponent rules.

Exponent Rules

Understanding exponent rules is crucial for simplifying algebraic expressions, especially those that involve powers. These rules allow you to manipulate expressions involving powers of the same base more easily. One of the fundamental rules is the power of a power rule: \((a^m)^n = a^{m \cdot n}\). This rule quickly simplifies terms like \((x^2)^3 = x^6\). Another important principle is the division of powers rule. When you divide like bases, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). This was used to transform \(\frac{2x^6}{3x^4}\) into \(\frac{2x^{6-4}}{3} = \frac{2x^2}{3}\). Remember, exponent rules streamline complex expressions and provide cleaner, more manageable solutions. They are powerful tools for algebra and frequently appear in calculus and beyond, forming the backbone of scientific calculations.

Numerator and Denominator

In mathematics, every fraction has two essential components: the numerator and the denominator. The numerator is the top part, indicating how many parts are in consideration, while the denominator is the bottom part, showing the total number of equal parts something is divided into. In our example, \(8x^6\) is the numerator, and \(12x^4\) is the denominator.Understanding the roles and relationships of these components is paramount, especially when simplifying fractions. Looking for common factors between the numerator and the denominator helps in reducing fractions. When you find these factors, you can divide both parts by them, which simplifies the expression without altering its value. Moreover, when variables are in both the numerator and denominator, use exponent rules to combine or simplify them. Here, simplifying relies on reducing both numerical parts (\(8\) and \(12\)) and cancelling out common \(x\) factors using exponent rules. This dual focus reduces \(\frac{8x^6}{12x^4}\) to \(\frac{2x^2}{3}\), offering a cleaner and simplified version of the expression.