Problem 85 - Intermediate Algebra

Question

Solve each equation. $$ (2 x-4)^{\frac{2}{3}}=1 $$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Exponentiated Expression

We start with the equation \((2x - 4)^{\frac{2}{3}} = 1\). Here, the term \((2x - 4)^{\frac{2}{3}}\) is already isolated, so we can proceed to the next operation.

Step 2: Eliminate the Fractional Exponent

To eliminate the fractional exponent, we raise both sides of the equation to the power of \(\frac{3}{2}\). This will remove the exponent on the left side: \[\left((2x - 4)^{\frac{2}{3}}\right)^{\frac{3}{2}} = 1^{\frac{3}{2}}\]\(2x - 4 = 1^{\frac{3}{2}}\) since \(1^{\frac{3}{2}} = 1\).

Step 3: Solve for x

With the equation \(2x - 4 = 1\), add 4 to both sides to isolate the terms containing \(x\): \[2x - 4 + 4 = 1 + 4\]\(2x = 5\). Now, divide both sides by 2 to solve for \(x\): \[x = \frac{5}{2}\].

Key Concepts

Solving Equations

An equation is essentially a balance scale. It consists of two equal expressions separated by an equals sign. Our goal when solving equations is to find the value of the unknown variable that satisfies this balance.

In the exercise given, we see the equation \((2x - 4)^{\frac{2}{3}} = 1\). Here, the expression \((2x - 4)^{\frac{2}{3}}\) is on one side, and the number 1 is on the other. To solve equations, we often need to manipulate them until we have the variable alone on one side. This can involve steps such as isolating terms, removing exponents, or simplifying expressions.

It's crucial to treat both sides of the equation equally. This means any operation performed on one side should also be applied to the other. When done correctly, this maintains the balance of the equation and guides us to the correct solution.

Fractional Exponents

Fractional exponents can seem intimidating, but they're just another way to express powers and roots. The equation \((2x - 4)^{\frac{2}{3}} = 1\) includes a fractional exponent, \(\frac{2}{3}\), which can be broken down into two parts:

• The numerator '2' - indicates a power.
• The denominator '3' - represents a root.

When dealing with fractional exponents, we can "unpack" them by raising the expression to the reciprocal of the fraction's exponent. In the exercise, raising both sides of the equation to the power of \(\frac{3}{2}\) cancels out the \(\frac{2}{3}\) exponent, simplifying our expression. Hence, knowing how to manage fractional exponents is essential for breaking down and solving equations effectively.

To feel more comfortable with fractional exponents, practice rewriting them as a combination of roots and powers. For example, \(x^{\frac{2}{3}}\) is equivalent to \((x^{2})^{\frac{1}{3}}\) or the cube root of \(x^{2}\). This technique helps simplify equations and reveals the underlying structure.

Algebraic Manipulation

In algebra, manipulation means rearranging or simplifying an equation into a more straightforward form. For the given exercise, algebraic manipulation played a crucial role in solving for \(x\). Once the fractional exponent was dealt with, the equation was reduced to a much simpler form: \(2x - 4 = 1\).

Now, it's time to apply basic algebraic manipulation to isolate \(x\). First, add 4 to both sides to cancel out the \(-4\) from the left side, resulting in \(2x = 5\). From here, dividing both sides by 2 isolates \(x\), giving the solution \(x = \frac{5}{2}\).

Breaking down problems into smaller, manageable steps is key in algebraic manipulation. Helpful strategies include:

• Adding or subtracting terms to both sides of the equation.
• Multiplying or dividing both sides by a non-zero number.
• Simplifying expressions before applying operations.

By understanding and applying these moves, solving complex equations like those with fractional exponents becomes much more approachable.