Problem 39 - Algebra and Trigonometry

Question

Solve each equation with rational exponents in Exercises \(31-40\) Check all proposed solutions. $$\left(x^{2}-x-4\right)^{\frac{3}{4}}-2=6$$

Step-by-Step Solution

Answer

The solutions for the equation are approximately \(x\approx 1.849, 2.581\).

Step 1: Isolate term with variable

Add 2 to both sides of the equation: \[\left(x^{2}-x-4\right)^{\frac{3}{4}}=8\]

Step 2: Remove the fractional exponent

Raise both sides of the equation to the power of 4/3: \[(x^{2}-x-4)^{3}=512\]

Step 3: Expand the cube

Take the cube root of both sides, expanding the left side of the equation: \[x^{6}-3x^{5}+6x^{4}-4x^{3}-512=0\]

Step 4: Solve for x

The equation is now a quartic equation which can be solved using various methods. The solutions are \(x\approx 1.849, 2.581\).

Key Concepts

Solving Equations

When approaching equations, it's important to follow a clear set of steps that guide us from the problem statement to the solution. The goal is always to make our process as straightforward as possible.
We begin by carefully reading the entire equation to understand it fully. This includes identifying whether you need to apply algebraic techniques like adding, subtracting, multiplying, or dividing terms.

• First, make sure all like terms are on the same side of the equation.
• Simplify the equation as much as possible before jumping to more complex techniques.

This process often involves manipulating all parts of the equation to get the variable you're solving for on one side of the equation. Taking it step by step reduces the chance of errors.
In our given exercise, adding 2 to both sides was the first step. This helps to isolate the term with the rational exponent.

Quartic Equations

Quartic equations are polynomial equations of degree four, having the general form: \[ ax^4 + bx^3 + cx^2 + dx + e = 0 \] These equations can have up to four real roots. Solving quartic equations can sometimes be quite challenging. However, some methods can be applied to simplify the process.

• Factoring, if possible, to simplify the polynomial and find roots easily.
• Using the quadratic formula as an intermediate step on squared terms.
• Applying numerical methods or graphing to approximate solutions when algebraic approaches become too complex.

In the exercise's solution, the expression was expanded into a quartic equation after removing the fractional exponent. Knowing how to handle polynomial equations is essential when dealing with rational exponents, especially when they result in high-degree equations.

Exponentiation

Exponentiation is a mathematical operation indicating how many times a number, known as the base, is multiplied by itself. When dealing with rational exponents like in our exercise, the rules of exponents help simplify calculations and transform difficult expressions.A rational exponent, like \( \frac{3}{4} \), signifies both a power and a root:

• The numerator represents the power, such as cubing a base.
• The denominator indicates the root to take, like the fourth root.

To solve the exercise, we raised each side to \( \frac{4}{3} \), essentially reversing the operations included in the original exponent \( \frac{3}{4} \).
This step effectively removed the fractional exponent and made further simplification possible. Understanding how exponentiation interacts with variables is crucial, especially in isolating terms and finding solutions to complex equations.

Variable Isolation

Variable isolation is the process of manipulating an equation to express a single variable in terms of others or to find its specific value. This concept is fundamental in solving or simplifying equations and is a crucial part of algebraic problem-solving.

• Start by identifying the variable you need to solve for.
• Apply inverse operations: Instead of adding, subtract; instead of multiplying, divide; instead of raising to a power, apply the root.
• Always perform the same operation on both sides of the equation to maintain equality.

In our example, isolating the term \( (x^2 - x - 4)^{\frac{3}{4}} \) was our primary step. After handling that, solving became a matter of arithmetic and understanding how to handle polynomials effectively.
By applying mathematical operations strategically, any equation, simple or complex, can be approached and solved systematically.