Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Find the real solution(s) of the equation involving rational exponents. Check your solutions. \((x+3)^{4 / 3}=16\)

Step-by-Step Solution

Answer

The solution of the equation \((x+3)^{4 / 3}=16\) is \(x = 5\).

Step 1: Isolate the term with the exponent

The original equation is \((x+3)^{4 / 3}=16\). The first step is to isolate the term with the rational exponent. This has been done in the original equation.

Step 2: Eliminate the fraction in the exponent

To eliminate the fraction in the exponent, raise both sides of the equation to the reciprocal of the fractional exponent. In this case, the reciprocal of \(4/3\) is \(3/4\). So, \(((x+3)^{4 / 3})^{3 / 4} = 16^{3 / 4}\). Simplifying this results in the equation \(x + 3 = 2√16 \rightarrow x + 3 = 8\).

Step 3: Solve for x

Finally, solve the resulting equation for x. Subtract 3 from both sides: \(x = 8 - 3\). Thus, \(x = 5\).

Step 4: Check the solution

Check the solution by substituting \(x = 5\), back into the original equation: \((5+3)^{4 / 3} = 16\). Simplifying the left side gives \(2^4 = 16\). Since both sides of the equation are equal, the solution is verified: \(x = 5\).

Key Concepts

Isolate the Term

In algebra, to solve equations effectively, we often start by isolating the term with the exponent or the variable of interest. This allows us to focus on the part of the equation that needs to be manipulated or solved. When we isolate a term, we're essentially separating it from the rest of the equation to deal with it individually.

For instance, if we have equation equation \((x+3)^{4/3}=16\), isolating the term with the exponent means we make sure that equation equation \((x+3)^{4/3}\) sits alone on one side of the equation. Luckily, in our given exercise, this step has already been done for us. If it hadn't been, we would need to move all other terms to the opposite side by performing inverse operations, such as subtracting a number from both sides of the equation, or dividing both sides by a coefficient.

Reciprocal of Fractional Exponent

Working with rational exponents can initially seem daunting, but understanding the concept of the reciprocal of a fractional exponent simplifies the process. A fractional exponent represents both an exponent and a root. The numerator indicates the power to which the number is raised, and the denominator signifies the root. For example, equation equation \(a^{m/n}\) implies taking the nth root of a, then raising it to the mth power.

When solving equations, we can get rid of a fractional exponent by raising the term to the reciprocal of the exponent. The reciprocal of a fraction is simply inverted; for fraction equation equation \(\frac{m}{n}\), the reciprocal is equation equation \(\frac{n}{m}\). Applying this to our equation equation equation \(((x+3)^{4/3})^{3/4}\), we use the reciprocal equation equation \(\frac{3}{4}\) of our original exponent equation equation \(\frac{4}{3}\) to remove the rational exponent, making it easier to solve the equation.

Checking Solutions in Algebra

After solving an equation, it's critical to check the solution to ensure it's correct. This involves substituting the found solution back into the original equation and verifying that the left and right sides equate correctly. When working with rational exponents or more complex expressions, this step is essential because it's easy to make mistakes with arithmetic operations or when manipulating the algebraic expressions.

For our exercise, we check the solution by substituting equation equation \(x=5\) back into equation equation \((x+3)^{4/3}=16\). Doing so should yield a true statement if we've solved the equation correctly. Simplifying equation equation \((5+3)^{4/3}\) results in equation equation \(2^4\), which is indeed equal to 16, confirming that our solution equation equation \(x=5\) is correct. Checking not only helps in verifying the solution but also reinforces understanding of manipulating equations with exponents.