Problem 36 - College Algebra Essentials

Question

Solve each equation with rational exponents. 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 \(x ≈ 2.2795\).

Step 1: Isolate the term with the exponent

The first step is to isolate the term with the exponent by adding 24 to both sides of the equation. This gives us \(8 x^{\frac{5}{3}} = 24\).

Step 2: Simplify the equation

You can simplify further by dividing both sides of the equation by 8. This will give \(x^{\frac{5}{3}} = 3\).

Step 3: Remove the rational exponent

Rational exponents represent both exponentiation and roots. \(x^{\frac{5}{3}}\) is equivalent to the cube root of \(x^5\). To remove the rational exponent, raise both sides to the power of \(\frac{3}{5}\). This results in \(x = 3^{\frac{3}{5}}\) which is the cube root of \(3^3\).

Step 4: Simplify the right side of the equation

Solving \(3^{\frac{3}{5}}\) gives us the final solution, which is approximately 2.2795 for \(x\).

Step 5: Checking the solution

Substitute \(x = 2.2795\) into the original equation to verify the solution. \(8*(2.2795)^{\frac{5}{3}} - 24\) should be approximately 0. If this equation holds true, then \(x = 2.2795\) is the correct solution.

Key Concepts

Isolate the Term

The process of solving equations often begins with isolating the term that contains the variable of interest. In the context of equations with rational exponents, this means separating the term with the exponent from other terms in the equation. This is a crucial step as it simplifies the equation and makes the subsequent steps more manageable.

For instance, considering the equation given in the exercise, the first necessary action is to add 24 to both sides, being careful to maintain the balance of the equation. This isolates the term with the exponent on one side, resulting in a cleaner, more straightforward equation where the focus is solely on the term with the variable exponent.

Rational Exponents

Rational exponents are exponents that are expressed as fractions. For many students, this concept can be perplexing because they involve both roots and powers. The key to understanding rational exponents is to recall that the numerator of the fraction is the power and the denominator is the root.

When encountering an expression like \(x^{\frac{5}{3}}\), it's essential to recognize that it represents the cube root of \(x^5\), due to the denominator being 3. This is critical because it informs how we can manipulate the term to eventually solve for \(x\).

Cube Root

The cube root is the operation of finding a number that, when raised to the third power, gives the original number. In mathematical terms, if \(y\) is the cube root of \(x\), then \(y^3 = x\).

In the equation from the exercise, once the rational exponent is interpreted as a cube root, you realize that you are essentially seeking the number which, when cubed, will yield our isolated term. To this end, exponentiation, specifically raising to the reciprocal of the fraction, is applied to remove the cube root, leaving the isolated variable.

Exponentiation

Exponentiation is the mathematical process of raising a number (the base) to the power of another number (the exponent). When dealing with rational exponents, exponentiation helps to 'undo' the root component of the exponent.

As demonstrated in the solution, by raising both sides of the equation \(x^{\frac{5}{3}} = 3\) to the reciprocal power of \(\frac{3}{5}\), the rational exponent is effectively eliminated, and the variable is solved. This inversion of the exponent's fraction is a powerful tool that facilitates the solving of equations that initially appear daunting due to the presence of rational exponents.