Simplify The Expression:$\[ \frac{x^3}{x^2-16} \div \frac{5x^3}{x^2-8x+16} \\]

Introduction

When it comes to simplifying complex expressions, one of the most challenging tasks is dividing rational expressions. Rational expressions are fractions that contain variables and constants in the numerator and denominator. Dividing these expressions can be a daunting task, but with the right approach, it can be simplified to a manageable form. In this article, we will explore the steps involved in simplifying the expression $\frac{x^3}{x^2-16} \div \frac{5x^3}{x^2-8x+16}$.

Understanding Rational Expressions

Before we dive into simplifying the expression, it's essential to understand what rational expressions are. A rational expression is a fraction that contains variables and constants in the numerator and denominator. Rational expressions can be simplified by canceling out common factors in the numerator and denominator. The key to simplifying rational expressions is to identify the common factors and cancel them out.

Simplifying the Expression

To simplify the expression $\frac{x^3}{x^2-16} \div \frac{5x^3}{x^2-8x+16}$, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Step 1: Factor the Denominators

The first step in simplifying the expression is to factor the denominators. The denominator of the first fraction is $x^2-16$, which can be factored as $(x+4)(x-4)$. The denominator of the second fraction is $x^2-8x+16$, which can be factored as $(x-4)(x-4)$.

Step 2: Rewrite the Expression

Now that we have factored the denominators, we can rewrite the expression as:

$$\frac{x^3}{(x+4)(x-4)} \div \frac{5x^3}{(x-4)(x-4)}$$

Step 3: Cancel Out Common Factors

The next step is to cancel out common factors in the numerator and denominator. We can cancel out the $(x-4)$ term in the numerator and denominator.

Step 4: Simplify the Expression

After canceling out the common factors, we are left with:

$$\frac{x^3}{(x+4)} \div \frac{5x^3}{(x-4)(x-4)}$$

Step 5: Invert and Multiply

To divide fractions, we need to invert the second fraction and multiply. Inverting the second fraction gives us:

$$\frac{(x-4)(x-4)}{5x^3}$$

Step 6: Multiply the Numerators and Denominators

Now that we have inverted the second fraction, we can multiply the numerators and denominators:

$$\frac{x^3(x-4)(x-4)}{(x+4)(x-4)(x-4)} \times \frac{(x-4)(x-4)}{5x^3}$$

Step 7: Cancel Out Common Factors

After multiplying the numerators and denominators, we can cancel out common factors. We can cancel out the $(x-4)$ term in the numerator and denominator.

Step 8: Simplify the Expression

After canceling out the common factors, we are left with:

$$\frac{x^3}{(x+4)} \times \frac{(x-4)}{5x^3}$$

Step 9: Cancel Out Common Factors

We can cancel out the $x^3$ term in the numerator and denominator.

Step 10: Simplify the Expression

After canceling out the common factors, we are left with:

$$\frac{(x-4)}{(x+4)} \times \frac{1}{5}$$

Step 11: Simplify the Expression

We can simplify the expression by multiplying the fractions:

$$\frac{(x-4)}{5(x+4)}$$

Conclusion

Simplifying the expression $\frac{x^3}{x^2-16} \div \frac{5x^3}{x^2-8x+16}$ requires a step-by-step approach. By following the order of operations and canceling out common factors, we can simplify the expression to a manageable form. The final simplified expression is $\frac{(x-4)}{5(x+4)}$.

Introduction

In our previous article, we explored the steps involved in simplifying the expression $\frac{x^3}{x^2-16} \div \frac{5x^3}{x^2-8x+16}$. We broke down the process into manageable steps, from factoring the denominators to canceling out common factors. In this article, we will answer some of the most frequently asked questions about simplifying rational expressions.

Q&A

Q: What is the first step in simplifying a rational expression?

A: The first step in simplifying a rational expression is to factor the denominators. This involves breaking down the denominator into its prime factors.

Q: How do I know if a rational expression can be simplified?

A: A rational expression can be simplified if there are common factors in the numerator and denominator. These common factors can be canceled out to simplify the expression.

Q: What is the difference between simplifying a rational expression and reducing a rational expression?

A: Simplifying a rational expression involves canceling out common factors, while reducing a rational expression involves canceling out any common factors and then simplifying the resulting expression.

Q: Can I simplify a rational expression with a zero denominator?

A: No, you cannot simplify a rational expression with a zero denominator. A zero denominator is undefined, and any expression with a zero denominator is considered to be undefined.

Q: How do I handle a rational expression with a negative exponent?

A: When a rational expression has a negative exponent, you can rewrite the expression with a positive exponent by taking the reciprocal of the expression.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you must be careful not to divide by zero.

Q: How do I know if a rational expression is in its simplest form?

A: A rational expression is in its simplest form if there are no common factors in the numerator and denominator that can be canceled out.

Q: Can I simplify a rational expression with a complex denominator?

A: Yes, you can simplify a rational expression with a complex denominator. However, you must be careful to factor the denominator correctly and cancel out any common factors.

Examples

Example 1: Simplifying a Rational Expression with a Zero Denominator

Simplify the expression $\frac{x^2}{x^2-4}$.

Solution: This expression cannot be simplified because the denominator is zero.

Example 2: Simplifying a Rational Expression with a Negative Exponent

Simplify the expression $\frac{1}{x^{-2}}$.

Solution: This expression can be simplified by taking the reciprocal of the expression: $\frac{x^2}{1}$.

Example 3: Simplifying a Rational Expression with a Variable in the Denominator

Simplify the expression $\frac{x^2}{x^2-4x}$.

Solution: This expression can be simplified by factoring the denominator: $\frac{x^2}{x(x-4)}$. We can then cancel out the common factor of $x$.

Conclusion

Simplifying rational expressions can be a challenging task, but with the right approach, it can be simplified to a manageable form. By following the steps outlined in this article and answering the frequently asked questions, you can simplify rational expressions with confidence. Remember to always factor the denominators, cancel out common factors, and be careful not to divide by zero.