Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \frac{\left(2 x^{2}\right)^{3}}{4 x^{4}} $$

Step-by-Step Solution

Answer

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

Step 1: Apply the Power to a Power Rule

First, apply the power to a power rule on the expression \((2x^2)^3\). This rule states \((a^m)^n = a^{m \times n}\). So we have: \((2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^{6}\).

Step 2: Simplify the Fraction

Now, substitute \(8x^6\) into the fraction: \(\frac{8x^6}{4x^4}\). Begin by simplifying the constants: \(\frac{8}{4} = 2\).

Step 3: Subtract the Exponents of Like Bases

For the variables, use the quotient rule of exponents which states \(\frac{a^m}{a^n} = a^{m-n}\). So, \(\frac{x^6}{x^4} = x^{6-4} = x^2\).

Step 4: Final Simplified Expression

Combine the results from steps 2 and 3 to get the final simplified expression: \(2x^2\). Therefore, \(\frac{8x^6}{4x^4} = 2x^2\).

Key Concepts

Power to a Power Rule

The Power to a Power Rule is a helpful tool when dealing with expressions in which a power is raised to another power. This rule is straightforward and vital for simplifying complex algebraic expressions. It states that when you have an expression of the form

• \((a^m)^n\), it simplifies to \(a^{m \times n}\).

To put it simply, you multiply the exponents. For instance, when looking at the expression

• \((2 x^2)^3\), both the constant and the variable are raised to the power of 3.

The constant 2 becomes \(2^3 = 8\), and since the exponent on \(x\) is 2, we multiply it by 3 to get \(x^{2 \times 3} = x^6\). Hence, the expression \((2 x^2)^3\) simplifies to \(8x^6\). Breaking it down in this way makes it easier to manage complex expressions.

Quotient Rule of Exponents

The Quotient Rule of Exponents is crucial when simplifying expressions with exponents divided by each other. This rule states that when you have a fraction like

• \(\frac{a^m}{a^n}\), it simplifies to \(a^{m-n}\).

This means you subtract the exponent in the denominator from the exponent in the numerator. Applying this rule is essential when simplifying algebraic fractions, as it allows for easy and organized solutions. For example, in the fraction

• \(\frac{8x^6}{4x^4}\), after dealing with the constants, you're left with \(\frac{x^6}{x^4}\).

Using the Quotient Rule, it simplifies to \(x^{6-4} = x^2\). It’s a simple yet powerful tool for quick simplification.

Algebraic Fraction Simplification

Algebraic Fraction Simplification often involves both constants and variables with exponents. The key to simplification is breaking down each part step by step. Begin by simplifying constants on top of each other. For instance,

• \(\frac{8}{4} = 2\).

Next, simplify the exponents using the Quotation Rule of Exponents as explained earlier. Once simplified, combine your results for a fully simplified expression. Taking our example, the expression

• \(\frac{8x^6}{4x^4}\) simplifies to \(\frac{8}{4} \cdot \frac{x^6}{x^4} = 2x^2\).

By following a structured approach, you can systematically simplify any algebraic fraction, making it easier to handle and understand. Remember to methodically handle constants and variables separately for clarity.