Problem 27 - Algebra 2

Question

Solve each equation. $$ (3 x-2)^{\frac{1}{5}}+6=5 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{3} \).

Step 1: Isolate the Radical Expression

Start by isolating the term with the fractional exponent. Subtract 6 from both sides to obtain \[ (3x-2)^{\frac{1}{5}} = 5 - 6 \] which simplifies to \[ (3x-2)^{\frac{1}{5}} = -1 \].

Step 2: Remove the Fractional Exponent

To eliminate the fractional exponent, raise both sides of the equation to the 5th power, giving \[ \left((3x-2)^{\frac{1}{5}}\right)^5 = (-1)^5 \]. This simplifies to \[ 3x-2 = -1 \].

Step 3: Solve for x

Add 2 to both sides to isolate the term with x: \[ 3x = -1 + 2 \], which simplifies to \[ 3x = 1 \]. Divide both sides by 3 to find \[ x = \frac{1}{3} \].

Key Concepts

Understanding Radical Equations

Radical equations involve mathematical expressions where a variable is under a root sign. In the context of the given exercise, we're dealing with a fifth root, as indicated by a fractional exponent. Solving these types of equations often requires isolating the radical expression first. This means you want to "get alone" the part of the equation which contains the radical. In our example, this was achieved by subtracting 6 from both sides of the equation. This step is crucial because it allows you to focus on simplifying the radical itself. Remember, whenever you work with radical equations, your goal is to eventually "free" the variable from the radical, so you can solve for it easily.

Exploring Fractional Exponents

Fractional exponents are another way to express roots. For example, the expression \( \sqrt[5]{3x-2} \) can be written as \((3x-2)^{\frac{1}{5}}\). These two notations are equivalent, but using fractional exponents often simplifies algebraic manipulation.

• The numerator of the fractional exponent indicates the power to which the base is raised.
• The denominator shows the root, making \(\frac{1}{5}\) represent the fifth root.

To solve equations with fractional exponents, it is critical to eliminate them by using the reciprocal of the fraction. In the example, raising both sides to the 5th power (the inverse of \(\frac{1}{5}\) ) removes the fractional exponent. This makes the equation much more straightforward to work with, converting the root into a simple linear expression.

Mastering Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations to solve for unknowns. In our exercise, we manipulated the equation in a few key ways. First, we isolated the radical expression by subtracting from both sides. This step simplifies the equation.Once the radical was isolated and removed with the power operation, we focused on the remaining linear equation.

• Adding numbers to both sides allowed us to further simplify the expression \(3x - 2 = -1\).
• Finally, dividing both sides by a constant gave the solution \(x = \frac{1}{3}\).

These basic algebraic operations are pillars in solving equations. They allow us to isolate the variable step by step. The goal is to work with the equation systematically, reducing it down to a simple statement where the variable is on one side, and its value is expressed on the other.