Problem 88 - Algebra for College Students

Question

Solve each equation. $$(4 x+5)^{\frac{2}{3}}=2$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{2\sqrt{2} - 5}{4}\).

Step 1: Isolate the Power

The given equation is \((4x + 5)^{\frac{2}{3}} = 2\). Start by isolating the expression with the fractional power, which is already done in this case.

Step 2: Eliminate the Fractional Power

Raise both sides to the power of \(\frac{3}{2}\) to eliminate the fractional exponent. By doing so, the equation becomes \((4x + 5) = 2^{\frac{3}{2}}\).

Step 3: Simplify the Power on the Right

Calculate \(2^{\frac{3}{2}}\). \(2^\frac{3}{2}\) can be rewritten as \((2^{\frac{1}{2}})^3 = \sqrt{2}^3 = \sqrt{8}\), which simplifies to \(2\sqrt{2}\).

Step 4: Solve for x

Now solve the equation \(4x + 5 = 2\sqrt{2}\). Subtract 5 from both sides to get \(4x = 2\sqrt{2} - 5\). Then, divide both sides by 4 to find \(x = \frac{2\sqrt{2} - 5}{4}\).

Key Concepts

Fractional Exponents

Fractional exponents can seem a bit intimidating at first. But, they can be understood by breaking down their components. When you see an exponent that has fractions, like \[(4x + 5)^{\frac{2}{3}}\], it is actually a compact way of expressing both an exponential operation and a root operation. Here’s how it works:

• The numerator (the number on top of the fraction) indicates the power to which the expression inside should be raised.
• The denominator (the number on the bottom) tells us what root to take.

So, \[(4x + 5)^{\frac{2}{3}}\] is essentially saying: first, raise \(4x + 5\) to the second power, and then take the cube root of that result. By understanding this, you can unravel expressions that otherwise might seem very complicated. Breaking down these steps is crucial to simplifying and solving them.

Isolation of Variables

The concept of isolating variables is foundational in solving equations like \((4x + 5)^{\frac{2}{3}} = 2\). The goal is to have the variable \(x\) alone on one side of the equation to find its value.Here’s a general approach:

• Identify the part of the equation containing the variable. In our case, it is \(4x + 5\).
• Apply operations that simplify the equation without altering its equality. This means what you do to one side, you must do to the other. For our example, we need to eliminate the fractional exponent. We do this by raising both sides of the equation to the power of the fraction's reciprocal, \(\frac{3}{2}\), resulting in \(4x + 5 = 2^{\frac{3}{2}}\).

Isolating variables in equations allows us to methodically solve for unknowns. Practicing this concept helps in solving more complex algebraic problems by breaking down the process into simpler operations.

Simplifying Radicals

Simplifying radicals, such as \(2^{\frac{3}{2}}\), is about expressing a radical in its simplest form. Let's break down the steps to simplify it:

• Convert the expression \(2^{\frac{3}{2}}\) by recognizing its components as taking a power and a root. Rewrite it as:\((2^\frac{1}{2})^3\).
• Recognize \(2^{\frac{1}{2}}\) as the square root, so we have \(\sqrt{2}^3\).
• Compute this as: \(\sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = \sqrt{8}\), which simplifies further to \(2\sqrt{2}\).

Combining powers and roots can initially appear challenging, but knowing how to simplify radicals is key for not only solving equations easily but also for recognizing patterns in algebra. This skill is especially useful when checking your work or when comparing results with others to see if they are indeed equivalent.