Problem 109 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In one short sentence, five words or less, explain what $$\frac{\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}}{\frac{1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}}$$ does to each number \(x\).

Step-by-Step Solution

Answer

Multiplies by \(x\).

Step 1: Simplification

The given expression is a complex fraction. Let's begin by simplifying it. Knowing the properties of exponents, we can rewrite the fractions in the numerator and denominator as powers of \(x\) as follows: \(\frac{x^{3}+x^{2}+x}{x^{2}+x+1}\).

Step 2: Finding the behavior of the expression

Now, by considering the highest power of \(x\) in the numerator and denominator, one can assess the behavior of the expression. Here, as \(x \rightarrow \infty\) or \(x \rightarrow -\infty\), the term with the highest power dominates the others in the binomial. So, for large \(x\), this expression can be approximated as \(\frac{x^{3}}{x^{2}} = x\).

Step 3: Describing the effect in less than five words

Knowing that the simplified form behaves like \(x\) for large \(x\), we can conclude that the effect of the expression on any number, \(x\), in one short sentence of five words or less is 'Multiplies by \(x\)'.

Key Concepts

Properties of Exponents

Understanding the properties of exponents can greatly simplify complex fractions. When dealing with fractions involving terms with exponents, remember the important rules:

• Product Rule: When multiplying like bases, add the exponents, e.g., \(x^a \times x^b = x^{a+b}\).
• Quotient Rule: When dividing like bases, subtract the exponents, e.g., \(\frac{x^a}{x^b} = x^{a-b}\).
• Power Rule: When raising a power to another power, multiply the exponents, e.g., \((x^a)^b = x^{a\times b}\).

In our expression \(\frac{\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}}{\frac{1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}}\), applying these rules allows us to rewrite and combine terms, making the complex fraction easier to handle.

Simplification

Simplifying a complex fraction like \(\frac{\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}}{\frac{1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}}\) involves rewriting it by using the properties of exponents. We accomplished this by transforming into a simpler form: \(\frac{x^{3}+x^{2}+x}{x^{2}+x+1}\).
First, notice each term of the original fraction is rewritten using the inverse of the power of \(x\). By using the reciprocal property, \(\frac{1}{x^a} = x^{-a}\), we're able to convert terms into multiplication-friendly expressions.
This transformation helps in combining fractions and eventually leads to a simpler, more manageable form, where the highest power terms dominate.

Behavior of Expressions

Analyzing the behavior of expressions is crucial to understanding how they function. In algebra, especially with rational expressions, behavior typically depends on variable values getting very small or very large.
To investigate the behavior of \(\frac{x^{3}+x^{2}+x}{x^{2}+x+1}\), focus on the highest power of \(x\) in both the numerator and the denominator. As \(x\) grows larger, these terms dominate, simplifying the expression to just the simplified power ratio.

• For our example: \(\frac{x^3}{x^2} = x\)

Identifying this helps us predict expression behavior under different conditions.

Asymptotic Behavior

Asymptotic behavior is a fascinating concept in understanding how functions behave as inputs become large or small. With rational expressions, the highest power terms dictate how the function behaves when approaching infinity or negative infinity.
In our expression, \(\frac{x^{3}}{x^{2}}\), the asymptotic behavior tells us that as \(x\) becomes very large (or very small), the expression approximately equals \(x\). This information reveals the simplified output when input values are extreme.

• This allows us to quickly deduce that the expression effectively 'Multiplies by \(x\)' for large values of \(x\).

Understanding this concept is vital for predicting how algebraic expressions relate to their variables in practical scenarios.