Problem 70 - Precalculus: Functions and Graphs

Question

Express as a sum of terms of the form \(a x^{\prime}\) where \(r\) is a rational number. $$\frac{x^{2}+4 x-6}{\sqrt{x}}$$

Step-by-Step Solution

Answer

The expression is \( x^{3/2} + 4x^{1/2} - 6x^{-1/2} \).

Step 1: Rewrite the Expression

The expression is \( \frac{x^2 + 4x - 6}{\sqrt{x}} \). To simplify this, recognize that \( \sqrt{x} \) is equivalent to \( x^{1/2} \). Substitute this into the expression to get \( \frac{x^2 + 4x - 6}{x^{1/2}} \).

Step 2: Divide Each Term

Now, divide each term in the numerator by \( x^{1/2} \). This gives:\[\frac{x^2}{x^{1/2}} + \frac{4x}{x^{1/2}} - \frac{6}{x^{1/2}}\]

Step 3: Simplify Each Term

Simplify each term by using the rules of exponents. For each term:1. \( \frac{x^2}{x^{1/2}} = x^{2 - 1/2} = x^{3/2} \)2. \( \frac{4x}{x^{1/2}} = 4x^{1 - 1/2} = 4x^{1/2} \)3. \( \frac{6}{x^{1/2}} = 6x^{-1/2} \) (note the negative exponent because x is in the denominator)This results in the expression: \( x^{3/2} + 4x^{1/2} - 6x^{-1/2} \).

Step 4: Express the Solution

We now express the expression as a sum of terms in the form \( a x^r \) where \( r \) is a rational number. The result is:\[ x^{3/2} + 4x^{1/2} - 6x^{-1/2} \]

Key Concepts

Rational Exponents

Rational exponents are an alternative way to express roots besides using radical signs, and they follow the rules of exponents. In the expression \( \sqrt{x} \), the square root of \( x \) can be rewritten as \( x^{1/2} \). Here, the exponent \( 1/2 \) is a rational number, as it is the quotient of two integers, 1 and 2.
Using rational exponents makes it easier to perform algebraic operations and simplifies expressions involving roots. For example

• When multiplying, the exponents are added, e.g., \( x^{a/b} \cdot x^{c/d} = x^{(ad+bc)/(bd)} \).
• When dividing, the exponents are subtracted, e.g., \( \frac{x^{a/b}}{x^{c/d}} = x^{(ad-bc)/(bd)} \).

This conversion may seem subtle, but it is crucial for simplifying expressions like in the original exercise.

Simplifying Expressions

Simplifying expressions is the process of making them easier to work with by reducing their complexity. It involves identifying and applying mathematical rules to transform an expression into a simpler but equivalent form. In the context of the problem, the main simplification comes from dividing each term in the numerator by the denominator which has a rational exponent.
Here are the steps for simplifying expressions with divisions:

• Identify common variables in the numerator and denominator.
• Use the properties of exponents to simplify each term individually.
• Apply the principle that a negative exponent indicates the reciprocal; for example, \( x^{-1/2} \) becomes \( 1/\sqrt{x} \).

Simplifying expressions correctly allows you to write them in forms that are easier to understand or work with during further calculations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They can be quite complex, but with knowledge of algebraic operations, they can be simplified or rewritten in a new form.
In the given exercise, understanding and rewriting algebraic expressions is key. This involves:

• Recognizing operations that combine terms, like addition or subtraction.
• Distributing division across terms, as seen when dividing each part of \( x^2 + 4x - 6 \) by \( x^{1/2} \).
• Substituting equivalent expressions, such as replacing radicals with rational exponents.

By mastering the manipulation of algebraic expressions, you can tackle various problems more confidently and effectively.