Problem 69 - Precalculus: Functions and Graphs

Question

Express as a sum of terms of the form \(a x^{\prime}\) where \(r\) is a rational number. $$\frac{4 x^{2}-x+5}{x^{2 / 3}}$$

Step-by-Step Solution

Answer

\(4x^{4/3} - x^{1/3} + 5x^{-2/3}\)

Step 1: Rewrite the Expression

Start with the given expression: \(\frac{4x^2 - x + 5}{x^{2/3}}\). Try to split it into separate fractions for each term in the numerator: \( \frac{4x^2}{x^{2/3}} - \frac{x}{x^{2/3}} + \frac{5}{x^{2/3}}\).

Step 2: Simplify Each Fraction

Simplify each fraction separately. For the first term, \( \frac{4x^2}{x^{2/3}} = 4x^{2 - 2/3} = 4x^{4/3}\). For the second term, \( \frac{x}{x^{2/3}} = x^{1 - 2/3} = x^{1/3}\). For the third term, \( \frac{5}{x^{2/3}} = 5x^{-2/3}\).

Step 3: Combine the Terms

Write all the simplified terms together as a sum: \(4x^{4/3} - x^{1/3} + 5x^{-2/3}\).

Key Concepts

Simplifying Expressions

Simplifying expressions can be likened to "tidying up" a mathematical phrase by following some precise steps. When tackling complex fractions like the given one, breaking it down into smaller pieces helps a lot. Each term can be simplified independently to create a new expression that is often easier to understand and work with. Here's how it can be done:

• Separation of terms: Begin by splitting the expression across the plus or minus signs, treating each division as a separate operation. This can turn one complicated fraction into several simpler ones, which is our first step.
• Simplify each fraction: Once separated, handle each fraction independently. For each fraction, the idea is to apply the basic algebraic rules, such as subtracting exponents, to simplify them to the simplest power possible. This is how you transition from power terms to more manageable forms.
• Recombination: After simplification of terms individually, you combine them again to form a single, simplified expression. This ultimately reveals the cleaner, more comprehensible form of the expression.

Breaking down complex math into bite-sized portions is key to grasping how and why these expressions can be simplified.

Rational Exponents

Rational exponents act as guidelines to express roots and powers in a unified way. Instead of sticking with traditional roots, like square or cube roots, rational exponents express these operations using fraction exponents that feature a numerator and a denominator.
Here’s a simple breakdown:

• Understanding the components: Any expression with a rational exponent such as \(x^{m/n}\) signifies the \(n\)-th root of \(x\) raised to the \(m\)-th power. Here, \(n\) serves as the root, and \(m\) as the power. So \(x^{2/3}\) translates to the cube root of \(x\) squared.
• Simplifying operations: Using the rule \(x^{a/b} \times x^{c/d} = x^{(ad+bc)/(bd)}\) and similar operations allows us to handle, combine, and simplify expressions effectively.

The beauty of rational exponents is they offer a smoother way to perform computations across varied forms of expressions, allowing a broad interplay of algebraic manipulations.

Polynomials

Polynomials are sturdy foundations within algebra. They consist of terms combined using addition, subtraction, and multiplication but not division by variables. They can consist of constants, variables, or a mix of both, where variables have whole number exponents.

• Terms of a polynomial: Each part of a polynomial separated by a plus or minus sign is known as a term. For example, \(4x^{2} - x + 5\) has three terms: \(4x^2\), \(-x\), and \(5\).
• Operations: Polynomial operations include addition, subtraction, multiplication, and division (divided directly, without remainder). Manipulating polynomials involves systematically applying these operations while respecting arithmetic rules.

Polynomials blend predictability with variety, providing a versatile avenue for simplifying expressions and solving equations, making them vital in further exploration of algebra and calculus.