Problem 49 - College Algebra with Modeling and Visualization

Question

Exercises \(45-54:\) Use the quotient rule to simplify the expression. Use positive exponents to write your answer. $$ \frac{24 x^{3}}{6 x} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4x^2 \).

Step 1: Apply the Quotient Rule for Exponents

The quotient rule states that when dividing two powers with the same base, you subtract the exponents. In this expression, we have \[ \frac{24 x^{3}}{6 x} \]where the base is \(x\). We subtract the exponents: \(3 - 1 = 2\).

Step 2: Simplify the Coefficients

Next, simplify the coefficients of the expression:\( \frac{24}{6} = 4 \).

Step 3: Write the Simplified Expression

Combine the simplified coefficient and the result from the exponent simplification to write the final simplified expression: \[ 4x^{2} \]

Key Concepts

Exponent Rules

Understanding the exponent rules is crucial in simplifying expressions like \(\frac{24 x^{3}}{6 x}\). The key rule to focus on here is the **Quotient Rule for Exponents**. This rule tells us that when dividing two expressions with the same base (in this case, the base is \(x)\), we simply subtract the exponent in the denominator from the exponent in the numerator. For example, with \(x^3\) over \(x^1\), we subtract the exponents:\[3 - 1 = 2\]. Thus, the expression becomes \(x^2\). Some important points to remember:

• The base must be the same for the quotient rule to apply.
• The quotient rule only affects the exponents, not the coefficients, which are handled separately.
• Always express the answer using positive exponents, which aligns with conventional mathematical practice.

Simplifying Expressions

Simplifying expressions may look complex at first, but by following the rules step by step, it becomes easy. When we apply both the quotient rule and arithmetic simplification, expressions become clearer and much simpler.First, apply the quotient rule to the exponents, as we discussed earlier. Next, deal with the coefficients—these are the numbers in front of the variables. In the original expression, it is important to simplify the coefficients as a separate step.For instance, in \(\frac{24 x^{3}}{6 x}\), we divide **24** by **6**, which gives us **4**. After simplifying both the exponents and the coefficients, we are left with \(4x^2\). Remember:

• Simplify exponents and coefficients separately for clarity.
• Use both arithmetic and exponent rules to reach the simplest form of the expression.

Algebraic Division

Algebraic division is a technique used to simplify expressions by dividing both the numbers and the variables in them. The process can seem tricky at first, but by breaking it down, it becomes manageable.In the expression \(\frac{24 x^{3}}{6 x}\):

• The first step involves dividing the coefficients: **24** divided by **6** equals **4**.
• The second step uses the quotient rule, which reduces the power of \(x\) by subtracting the exponents, leaving us with \(x^2\).
• Combine these results to get the simplified expression of \(4x^2\).

This procedure ensures we simplify all aspects of the expression methodically. By doing so, algebraic division makes handling complex expressions less daunting and more intuitive.