Problem 38 - Algebra and Trigonometry

Question

Solve each equation with rational exponents in Exercises \(31-40\) Check all proposed solutions. $$(x+5)^{\frac{2}{3}}=4$$

Step-by-Step Solution

Answer

After performing all the steps, we find that \(x = 3\).

Step 1: Remove the Rational Exponent

To get rid of the rational exponent, raise both sides of the equation to the power that is the reciprocal (the flipped version) of the rational exponent. Here, the exponent is \(\frac{2}{3}\), so its reciprocal will be \(\frac{3}{2}\). So, \((x+5)^{\frac{2}{3} * \frac{3}{2}} = 4^{\frac{3}{2}}\).

Step 2: Simplify

After step 1, we will have \(x+5 = 4^{\frac{3}{2}}\). Simplifying \(4^{\frac{3}{2}}\) yields \(8\). The equation becomes \(x+5 = 8\) after simplification.

Step 3: Solve for \(x\)

Finally, to find \(x\), we subtract \(5\) from both sides of the equation. This leaves us with \(x = 8 - 5\).

Key Concepts

Solving Equations with Rational Exponents

Solving equations that feature rational exponents can initially seem daunting, but once you understand the process, it becomes much more manageable. The key to solving such equations is to eliminate the rational exponent, which simplifies the equation significantly. A rational exponent is simply a way of expressing roots and powers in terms of fractions. For example, the expression \((x+5)^{\frac{2}{3}}\) indicates that we are dealing with a cube root raised to the second power.

To isolate the variable, you need to "undo" the power. This is done by raising both sides of the equation to the reciprocal of the original rational exponent. In this exercise, the exponent \(\frac{2}{3}\) is transformed by using its reciprocal, \(\frac{3}{2}\). By applying this logic, the left-hand side of the equation reduces to just \(x+5\), since the reciprocals cancel each other out. The equation becomes easier to solve as a result. Thus, knowing how to work with reciprocals is invaluable when tackling such problems.

After getting rid of the rational exponent, the equation becomes straightforward. Now, the task is to complete elementary operations to find the value of \(x\). Remember, practice makes perfect, and understanding these steps will aid greatly in mastering equations with rational exponents.

Understanding Exponent Rules

The realm of exponents is vast, and understanding its rules is crucial for solving equations efficiently. One handy rule is that of working with powers of powers, which is expressed mathematically as \((a^{m})^{n} = a^{m \cdot n}\). This is what allows us to simplify expressions such as \((x+5)^{\frac{2}{3} \cdot \frac{3}{2}}\). When you multiply these fractions, you get 1, meaning the result is simply the base itself, \(x+5\).

It's also helpful to recall the rule concerning fractional exponents: a fraction \(\frac{m}{n}\) as an exponent suggests an \(n\)-th root raised to the \(m\)-th power. So, \(4^{\frac{3}{2}}\) essentially means "the square root of \(4\), raised to the power of 3." First, calculate the square root of 4, which is 2, and then cube it, resulting in 8.

With these rules, manipulating and simplifying complex expressions becomes less of a chore and more of a process of applying logical mathematical operations. Make sure to work through each rule slowly and steadily for better comprehension and retention.

Using Reciprocal Exponents for Solving Equations

Reciprocal exponents are key to unlocking solutions in equations involving powers. Think of a reciprocal as simply flipping a fraction. When confronted with a fractional exponent like \(\frac{2}{3}\), using its reciprocal, \(\frac{3}{2}\), is your ticket to simplification.

By raising both sides of the equation to this power, you effectively eliminate the rational exponent on \((x+5)\). The base becomes neatly isolated, allowing you to address the equation much more directly. Consequently, the process brings us from \((x+5)^{\frac{2}{3}} = 4\) to simply \(x+5 = 4^{\frac{3}{2}}\).

This method is not just a trick but a principle grounded in the law of exponents, where multiplying an exponent by its reciprocal results in 1. The final step usually involves solving the newly simplified equation using basic arithmetic operations, finding \(x\) by subtracting constants from both sides. Reciprocal exponents thus empower you to transform and solve what might seem like complicated equations.