Problem 75 - Precalculus: Functions and Graphs

Question

Express as a quotient. $$x^{-1 / 2}-x^{3 / 2}$$

Step-by-Step Solution

Answer

Quotient is \( \frac{1 - x^2}{x^{1/2}} \).

Step 1: Identify the expression

The expression given is \( x^{-1 / 2} - x^{3 / 2} \). Our task is to express this as a quotient.

Step 2: Apply Negative Exponent Rule

The term \( x^{-1/2} \) can be rewritten using the negative exponent rule \( a^{-b} = \frac{1}{a^b} \). Hence, \( x^{-1/2} \) becomes \( \frac{1}{x^{1/2}} \).

Step 3: Apply the concept of common denominators

To combine terms into a single fraction, identify a common denominator for both terms. The denominators here are \( x^{1/2} \) and 1, so the common denominator is \( x^{1/2} \).

Step 4: Combine terms into a single fraction

Rewrite the expression \( x^{-1 / 2} - x^{3 / 2} \) using the common denominator. The expression becomes \( \frac{1}{x^{1/2}} - \frac{x^{3/2} \cdot x^{1/2}}{x^{1/2}} \). Simplify this to \( \frac{1 - x^2}{x^{1/2}} \).

Step 5: Final expression as a quotient

The final expression in the form of a quotient is \( \frac{1 - x^2}{x^{1/2}} \).

Key Concepts

Negative Exponents

Understanding negative exponents can simplify complex expressions. Exponents indicate how many times a number, known as the base, is multiplied by itself. A negative exponent, like in \( x^{-1/2} \), tells us to take the reciprocal of the base raised to the positive of that exponent.
For instance, the expression \( x^{-1/2} \) can be rewritten as \( \frac{1}{x^{1/2}} \). In this format, it's clear we are dealing with a division operation once inverted. It essentially flips the term, helping transform the expression into a more usable form for further processing.

Common Denominators

Having a common denominator is crucial when working with fractions or rational expressions. It allows us to combine multiple terms into a single fraction.
In algebra, finding a common denominator often requires identifying the least common multiple of the denominators. In our exercise, the terms \( \frac{1}{x^{1/2}} \) and \(-x^{3/2} \) can be combined by using \( x^{1/2} \) as a common denominator.
This step centralizes the expression making it easier to perform operations like addition, subtraction, or further simplification.

Simplifying Expressions

Simplifying expressions is a key algebraic skill that involves reducing fractions or combining terms into their simplest form. It often involves the use of common denominators, factoring, and simplifying exponents.
In the given problem, after finding a common denominator, the expression \( \frac{1}{x^{1/2}} - \frac{x^{3/2}\cdot x^{1/2}}{x^{1/2}} \) is simplified by combining like terms.
The expression simplifies to \( \frac{1 - x^2}{x^{1/2}} \), which is neater and more manageable for further calculations if needed.

Algebraic Fractions

Algebraic fractions are fractions that contain algebraic expressions in their numerators, denominators, or both. Understanding how to manipulate them is vital in algebra.
They behave like numerical fractions but involve additional operations such as distributing exponents across terms and factoring.
The exercise provided uses an algebraic fraction to express the quotient \( \frac{1 - x^2}{x^{1/2}} \). Managing algebraic fractions involves ensuring the expression is simplified and all operations are accurately performed, keeping track of variables and exponents.