Problem 87 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The given equation is \((2x-4)^{\frac{2}{3}}=1\). We need to isolate and solve for \(x\). This involves removing the fractional exponent by raising both sides to the reciprocal of the exponent.

Step 2: Eliminate the Fractional Exponent

Raise both sides of the equation to the power of \(\frac{3}{2}\) to eliminate the fractional exponent. This yields:\[\left((2x-4)^{\frac{2}{3}}\right)^{\frac{3}{2}} = 1^{\frac{3}{2}}\]This simplifies to:\[2x-4 = 1\]

Step 3: Solve for x

Isolate \(x\) by first adding 4 to both sides of the equation:\[2x-4+4 = 1+4\]This simplifies to:\[2x = 5\]Now, divide both sides by 2:\[x = \frac{5}{2}\]

Step 4: Verify the Solution

Substitute \(x = \frac{5}{2}\) back into the original equation to ensure it satisfies the equation:\[(2 \times \frac{5}{2} - 4)^{\frac{2}{3}} = (5 - 4)^{\frac{2}{3}} = 1^{\frac{2}{3}} = 1\]Since both sides are equal, \(x = \frac{5}{2}\) is confirmed as the correct solution.

Key Concepts

Fractional Exponents

Fractional exponents might seem tricky at first, but they are just another form of expressing roots and powers together in one neat package. When you see an exponent like \( \frac{2}{3} \), it means you are dealing with both a square and a cube root.
The numerator (\(2\) in this case) tells you to square the number or expression, while the denominator (\(3\)) indicates you take the cube root. Hence, \((2x-4)^{\frac{2}{3}}\) means take \((2x-4)\), square it, and then take the cube root of the result.

• Numerator as power: Think of the numerator as the power the base is raised to.
• Denominator as root: Consider the denominator as the type of root to be applied.
• Interpreting expressions: Applying fractional exponents means you might do operations in two steps: first root, then power, or vice versa.

Understanding fractional exponents is crucial for simplifying and solving equations efficiently.

Equation Manipulation

Equation manipulation involves using algebraic techniques to isolate the variable you're solving for. In our case, \( (2x-4)^{\frac{2}{3}}=1 \), we need to work through several manipulation steps to find \( x \).
First, to tackle fractional exponents, we used reciprocal powers. By raising both sides of the equation to \( \frac{3}{2} \), we effectively canceled out the fractional exponent. This is because multiplying \( \frac{2}{3} \) by \( \frac{3}{2} \) gives \( 1 \), leaving the base expression untouched.

• Reciprocal of Exponents: The reciprocal exponent method simplifies the equation first.
• Isolation of Variable: Adjust the equation step-by-step until the variable is alone.
• Operations Both Sides: Remember to perform the same operation on both sides to maintain equation balance.

Keeping an eye on keeping equations balanced is key, ensuring every operation is mirrored on the left and right sides.

Verification of Solutions

After finding a potential solution, always perform a verification step to confirm its validity. Here, we found \( x = \frac{5}{2} \) and substituted it back into the original equation to ensure no errors were made.
Substitute \( x = \frac{5}{2} \) into the expression \( (2x-4)^{\frac{2}{3}} \) to see if it equals \( 1 \). If both sides of the equation match, this confirms the solution.

• Re-substitution: Always plug the solution back into the original equation.
• Check for Equality: Validate if both sides result in the same value.
• Error Check: This step catches any previous mishaps in calculation or logic.

Verification is critical as it serves as a final check of your work, ensuring correctness.