Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Find the real solution(s) of the equation involving rational exponents. Check your solutions. \((x-5)^{2 / 3}=16\)

Step-by-Step Solution

Answer

The real solution for the equation \((x-5)^{2 / 3}=16\) is \(x = 261\).

Step 1: Simplify the equation

First, get rid of the fraction 2/3 by taking both sides of the equation to the power of \(3/2\). This gives us \(((x-5)^{2 / 3})^{3 / 2}=16^{3 / 2}\). Which simplifies to \((x-5)^{1}=256\).

Step 2: Solving for \(x\)

In this step, isolate \(x\) by adding 5 to both sides of the equation. This gives us \(x = 256 + 5\).

Step 3: Computing the solution

Evaluate the expression to find the solution. This gives \(x = 261\).

Step 4: Checking the solution

To verify the solution is correct, substitute \(x = 261\) back into the original problem \((x-5)^{2 / 3}=16\). If the left-hand side equals to the right-hand side, then the solution is correct. This gives \((261-5)^{2 / 3}=16\). After calculating, we get \(16=16\) which satisfies the original equation thus verifying \(x = 261\) to be the correct solution.

Key Concepts

Real Solutions

In mathematics, real solutions are the values that satisfy an equation when considering only real numbers. When tackling equations that contain rational exponents, or any mathematical problem for that matter, real solutions represent the answers that make an equation true within the set of real numbers.

• Real numbers encompass all the numbers you can think of in real-world scenarios – whole numbers, fractions, decimals, and negatives.
• Finding the real solutions to an equation is a pivotal part of verifying the relevance of the solution in real-world contexts.

Here, we are solving the equation \((x-5)^{2/3} = 16\) to find the real values of \(x\) that satisfy this condition. Through this process, identifying real solutions ensures that we avoid complex or imaginary numbers which wouldn't be applicable in scenarios constrained to the real number system.

Verifying Solutions

Verification is the process of confirming that a proposed solution actually satisfies the equation in question. Once you have derived a potential solution, such as \(x = 261\) in our example, it's crucial to check this value by plugging it back into the original equation.
To verify:

• Substitute the calculated solution back into the original equation: \((x-5)^{2/3} = 16\).
• Calculate the left-hand side to see if it equals the right-hand side. If these values match, the solution is verified.

For our equation, substituting \(x = 261\) results in \((261-5)^{2/3} = 16\), confirming the correctness of the solution with both sides equating to 16. Verification ensures both accuracy and clarity, validating that the path taken during solving was correctly executed.

Rational Exponents

Rational exponents represent powers that are fractions, which adds layers of complexity to solving equations. In \((x-5)^{2/3} = 16\), the exponent \(2/3\) suggests a series of operations — a power and a root combined:

• The numerator (2) indicates the exponent applied to the base \((x-5)\).
• The denominator (3) represents a root; specifically, this denotes the cube root in computations over the term.

This form allows us to operationalize different ways to solve the equation, like using power properties to simplify or invert exponents to isolate terms. Understanding rational exponents can be crucial for manipulating and simplifying expressions in mathematics at various levels.

Step-by-Step Solution

Approaching any equation with a step-by-step solution process brings structure and clarity to solving mathematical problems.
For the equation \((x-5)^{2/3} = 16\), breaking it down involves:
1. **Removing the Rational Exponent**: In our example, we took each side to the \((3/2)\) power to eliminate the fraction exponent, known as rationalizing the term.
2. **Isolating \(x\)**: After simplifying the equation to \((x-5)^{1} = 256\), the next step is solving for the variable \(x\) by isolating it on one side of the equation.
3. **Computing the Solution**: Evaluate the expression to arrive at a numerical solution (\(x = 261\) in this case).
4. **Verification**: Always end by substituting back to ensure the solution works in the original context.
This organized approach ensures no steps are overlooked, and helps in systematically solving often complex mathematical relationships.