Problem 14 - Precalculus: Functions and Graphs

Question

Simplify. $$\frac{\left(2 x^{2}\right)^{3}}{4 x^{4}}$$

Step-by-Step Solution

Answer

The simplified form is \(2x^2\).

Step 1: Expand the Numerator

Begin by expanding the expression in the numerator. The term \((2x^2)^3\) indicates that we need to cube both the constant \(2\) and the variable part \(x^2\). This results in: \[(2^3)\cdot(x^{2\cdot3}) = 8x^6\].

Step 2: Write the Full Expression

Substitute the expanded form of the numerator back into the original expression to get: \[\frac{8x^6}{4x^4}\].

Step 3: Simplify the Coefficient

Divide the numerical coefficients 8 by 4: \[\frac{8}{4} = 2\]. This simplifies the expression to: \[\frac{2x^6}{x^4}\].

Step 4: Simplify the Exponents

Apply the rule of subtraction for exponents when dividing like bases: \[x^6 \div x^4 = x^{6-4} = x^2\].

Step 5: Write the Simplified Expression

Combine insights from previous steps to arrive at the simplified form: \[2x^2\].

Key Concepts

Understanding Exponent Rules

Exponents are a shorthand notation for expressing repeated multiplication of a number by itself. Let's simplify how they work. When you see an expression like

• \((a^m)^n\), you apply the rule to multiply the exponents together: \(a^{m \cdot n}\).
• For multiplication with the same base, you'll add the exponents: \(x^m \cdot x^n = x^{m+n}\).
• For division with the same base, you subtract the exponents: \(\frac{x^m}{x^n} = x^{m-n}\).

When you simplify operations involving powers of numbers and variables, these rules let you transform a complex expression into a much simpler one. In our problem, applying \((2x^2)^3 = 8x^6\) used the exponent rule to expand and simplify the numerator.

The Art of Algebraic Simplification

Algebraic simplification reduces an expression to its simplest form, making complex work easier. It involves a step-by-step reduction of terms. Start by identifying similar terms that can be combined or cancelled.

• You look for opportunities to factor numbers and variables when possible.
• Simplifying coefficients, such as dividing \(\frac{8}{4} = 2\), helps reduce complexity.

Simplification in algebra is a balancing act of precision and skills. You identify the core components of expressions, streamline them to more manageable forms, and apply consistent mathematical techniques to guide you to the simplest possible solution. This way, we transformed \(\frac{8x^6}{x^4}\) into \(2x^2\) by simplifying both the coefficient and the exponents using appropriate rules.

Navigating Rational Expressions

Rational expressions are fractions in which the numerator and/or the denominator are polynomials. Simplifying these expressions is similar to simplifying numeric fractions. Follow these steps:

• First, expand and simplify both the numerator and the denominator, if needed.
• Look for common factors that can be cancelled. Remove commonalities following algebraic simplification steps.
• Simplify the coefficients and apply the exponent rules to the variables.

In the exercise, we simplified the entire expression \(\frac{(2x^2)^3}{4x^4}\) by working on the numerator and denominator separately. We expanded the numerator to \(8x^6\) and simplified the result with the denominator to arrive at \(2x^2\). Handling rational expressions with these steps ensures clarity and precision in simplification.