Problem 54 - Beginning and Intermediate Algebra

Question

Solve. $$\sqrt[5]{2 x-3}-1=1$$

Step-by-Step Solution

Answer

The solution to the given equation \(\sqrt[5]{2x-3}-1=1\) is \(x = \frac{35}{2}\).

Step 1: Isolate the 5th root term

We want to isolate the term with the 5th root: \(\sqrt[5]{2x-3}\). To do so, we need to add 1 to both sides of the equation: \[\sqrt[5]{2x-3} - 1 + 1 = 1 + 1\] \[\sqrt[5]{2x-3} = 2\]

Step 2: Eliminate the 5th root

To eliminate the 5th root, we will raise both sides of the equation to the power of 5: \[(\sqrt[5]{2x-3})^5 = 2^5\] By doing this, the left side simplifies to: \[2x-3 = 2^5\]

Step 3: Solve the resulting equation

Now we have a simpler equation to solve, \(2x-3 = 2^5\). First, we will calculate \(2^5\): \[2x-3 = 32\] Next, add 3 to both sides of the equation: \[2x = 35\] Finally, divide both sides by 2: \[x = \frac{35}{2}\] So the solution to the given equation is \(x = \frac{35}{2}\).

Key Concepts

Fifth Roots

In mathematics, a fifth root of a number is a specific instance of taking the root of a number. It is the number that, when multiplied by itself five times, gives you the original value. For an equation containing a fifth root, such as \(\sqrt[5]{2x-3}\), our goal is often to isolate and eliminate the root to solve for the variable. Understanding fifth roots involves:

• Recognizing they work similarly to square or cube roots but require multiplication five times.
• Understanding that \(\sqrt[5]{a}\) signifies the number that gives "a" when raised to the power of 5.
• Realizing that if \(\sqrt[5]{b} = c\), then \(b = c^5\).

Grasping these concepts is crucial for manipulating and solving radical equations effectively.

Isolation of Terms

Isolating a term in an equation means rearranging the equation so that the term of interest is by itself on one side. In the given exercise, we start with \(\sqrt[5]{2x-3} - 1 = 1\).

• To isolate the fifth root, we first add 1 to both sides, resulting in \(\sqrt[5]{2x-3} = 2\).
• This step is essential because it prepares the equation for eliminating the root by making the equation simpler.

Successful isolation ensures clearer manipulation in subsequent steps, making it easier to apply further operations to solve the equation.

Raising to a Power

Raising both sides of an equation to a power can help eliminate roots and simplify the equation. It's a powerful tool used to transform radical equations into polynomial forms.

• When we have \(\sqrt[5]{2x-3} = 2\), raising both sides to the power of 5 eliminates the fifth root.
• This results in \((\sqrt[5]{2x-3})^5 = 2^5\), simplifying to \(2x-3 = 32\).
• Raising to a power should be done carefully to ensure that the operations are valid across all possibilities for the variable.

This technique allows us to work with simpler algebraic expressions, facilitating easier problem solving.

Simplifying Equations

Simplifying equations involves performing operations to make the equation easier to solve and understand. After raising both sides to the fifth power, we're left with a linear equation that is straightforward in form: \(2x-3 = 32\).To solve this:

• Add 3 to both sides to get \(2x = 35\).
• Divide both sides by 2 to isolate \(x\), which gives \(x = \frac{35}{2}\).

These operations are simple arithmetic steps that result in the final solution. Simplification leads to clarity, making complex problems manageable and solutions attainable.