Problem 72 - Precalculus: Functions and Graphs

Question

Express as a sum of terms of the form \(a x^{\prime}\) where \(r\) is a rational number. $$\frac{(\sqrt{x}-3)^{2}}{x^{3}}$$

Step-by-Step Solution

Answer

\(x^{-2} - 6x^{-5/2} + 9x^{-3}\).

Step 1: Expand the Numerator

We need to expand the expression \((\sqrt{x} - 3)^2\) which comes out to \((\sqrt{x})^2 - 2\cdot\sqrt{x}\cdot3 + 3^2\). Simplifying, we get \(x - 6\sqrt{x} + 9\). Thus, the numerator of the given expression becomes \(x - 6\sqrt{x} + 9\).

Step 2: Divide Each Term by the Denominator

We will now divide the expanded numerator term by term by the denominator \(x^3\). 1. First term: \(\frac{x}{x^3} = x^{-2}\).2. Second term: \(\frac{-6\sqrt{x}}{x^3}\). Since \(\sqrt{x} = x^{1/2}\), it simplifies to \(-6x^{-5/2}\).3. Third term: \(\frac{9}{x^3} = 9x^{-3}\).

Step 3: Express Final Result

Combine the terms divided by the denominator from Step 2 into a single expression:- The expression is now \(x^{-2} - 6x^{-5/2} + 9x^{-3}\). - Each term is in the form \(ax^r\), where \(r\) is a rational number.

Key Concepts

Rational Exponents

Rational exponents are a way to express powers and roots in a unified manner. Understanding them helps simplify complex expressions, especially in polynomial division problems. When dealing with rational exponents, you express a root as a fractional power. For instance, the square root of a number can be written with an exponent of 1/2. This is particularly useful because it allows for the use of rules of exponents to simplify or manipulate expressions.

Let's take a look at how rational exponents work with the original exercise. In this problem, you are given \(\sqrt{x}\), which is equivalent to \(x^{1/2}\). By rewriting roots as exponents, we can more easily manipulate the terms using exponent rules:

• Multiplying with a Same Base: Add the exponents, for example, \(x^{m} \cdot x^{n} = x^{m+n}\).
• Dividing with a Same Base: Subtract the exponents, such as \(\frac{x^{m}}{x^{n}} = x^{m-n}\).
• Power of a Power: Multiply the exponents, like \((x^{m})^{n} = x^{m \cdot n}\).

This conversion is crucial when you perform polynomial division, and it simplifies the handling of the expression.

Expansion of Binomials

Expanding expressions, especially binomials, is a common task in algebra. The original expression \((\sqrt{x} - 3)^{2}\) is a binomial. To expand it, you use a process called 'expanding binomials'.

To expand a binomial of the form \((a - b)^{2}\), we apply the formula:
\[(a - b)^2 = a^2 - 2ab + b^2\]

This formula helps us quickly expand any binomial without lengthy multiplication processes, translating in our original problem to \(x - 6\sqrt{x} + 9\). Breaking it further down, we did:

• First term's square: \((\sqrt{x})^{2} = x\)
• Twice the product of both terms: \(-2 \cdot \sqrt{x} \cdot 3 = -6\sqrt{x}\)
• Second term's square: \(3^{2} = 9\)

This step is essential for simplifying expressions further and prepares the terms for division.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. In this problem, once you have expanded the binomial, the goal is to simplify it by polynomial division. Simplifying makes it easier to work with expressions, and it's a valuable skill for solving algebraic equations.

In the original problem, each term of the expanded numerator is divided by the denominator \(x^{3}\). Here are the steps:

• First Term: \(\frac{x}{x^3} = x^{-2}\) by subtracting the exponents.
• Second Term: \(-6\frac{\sqrt{x}}{x^3}\) transforms to \(-6x^{-5/2}\) using the rule \(x^{1/2 - 3} = x^{-5/2}\).
• Third Term: \(\frac{9}{x^3} = 9x^{-3}\), already simplified as a rational exponent.

Combining these terms, the expression \(x^{-2} - 6x^{-5/2} + 9x^{-3}\) is the simplified form. This form emphasizes the fundamental polynomial division technique, applying it step by step ensures no errors and results in an easily interpretable format.