Problem 36 - Algebra and Trigonometry

Question

Solve each equation with rational exponents in Exercises \(31-40\) Check all proposed solutions. $$8 x^{\frac{5}{3}}-24=0$$

Step-by-Step Solution

Answer

The solution to the equation \(8 x^{\frac{5}{3}}-24=0\) is \(x \approx 2.28 \)

Step 1: Isolate Term with Variable

First isolate the term with the variable on one side of the equation: \[8 x^{\frac{5}{3}}= 24\]

Step 2: Divide by the Coefficient

Then divide both sides by the coefficient, 8\[x^{\frac{5}{3}}= 3\]

Step 3: Apply the Reciprocal Exponent

Raising both sides to the power of the reciprocal of \(\frac{5}{3}\) which is \(\frac{3}{5}\) will remove the rational exponent on x:\[x=(3)^{\frac{3}{5}}\]

Step 4: Evaluate the Expression

Then, evaluate the right side of the equation to calculate the value of x:\[x\approx 2.28\]

Step 5: Check the Solution

Substitute x=2.28 into the original equation to check if it is satisfied:\[8(2.28)^{\frac{5}{3}}-24 \approx 0 \] Since the left side is approximately equal to the right side '0', the solution is correct.

Key Concepts

Rational Exponents

Understanding rational exponents is essential when solving equations involving powers and roots. A rational exponent, such as \( \frac{5}{3} \) found in the exercise, represents a number that is both a power and a root. To clarify, \( x^{\frac{5}{3}} \) is the same as \( \sqrt[3]{x^5} \) or the cube root of \( x \) raised to the fifth power. It is important to remember that the numerator in a rational exponent is the power to which the base is raised and the denominator is the root being taken. Thus, manipulating rational exponents requires understanding of both exponentiation and root extraction.

In the solved problem, the exponent \( \frac{5}{3} \) needs to be dealt with by applying a reciprocal exponent, another concept critical to finding the solution, as it eliminates the fractional power from the variable.

Isolating Variables

To solve for a variable means to get that variable by itself on one side of an equation. This process, called isolating the variable, allows you to clearly see what the variable equals without any additional terms or coefficients in the way. The first step in the provided exercise involves isolating \( x^{\frac{5}{3}} \) by getting rid of the '-24' on one side of the equation. The essential steps generally include using addition or subtraction to move terms and division or multiplication to deal with coefficients.

For more complex equations involving rational exponents, isolation of the variable might require additional algebraic manipulations such as taking roots or applying reciprocal exponents.

Reciprocal Exponents

The reciprocal of an exponent is simply flipping it over. If you have an exponent like \( \frac{5}{3} \) the reciprocal exponent is \( \frac{3}{5} \) since the reciprocal of a fraction is made by swapping the numerator and the denominator. In the context of solving equations with rational exponents, we use the property that a number raised to an exponent and then raised to a reciprocal exponent equals the base number to the power of one, or simply the base number itself.

This is why in step 3, raising both sides of the equation \( x^{\frac{5}{3}} = 3 \) to the power of \( \frac{3}{5} \) eliminates the complex exponent, simplifying the process of finding \( x \) and bringing us one step closer to the solution.

Equation Checking

Checking your solutions is a critical step in solving any algebraic equation, and it becomes even more important when dealing with rational exponents. To verify the solution, plug the value of \( x \) back into the original equation and ensure both sides balance. In this case, we substitute \( x \) with the approximate value of 2.28 and aim for the left side to have a value close to zero.

Because we are often dealing with approximate values, especially when roots are involved, it’s common for the result not to be exact but close enough to confirm the solution is correct. Therefore, equation checking serves as a means to validate the steps taken and the correctness of the final answer.