Problem 83 - Intermediate Algebra

Question

Solve each equation. $$ x^{\frac{2}{5}}=2 $$

Step-by-Step Solution

Answer

\( x = 4\sqrt{2} \)

Step 1: Isolate the Variable

We begin by isolating the expression that contains the variable, which is already done here: \( x^{\frac{2}{5}} = 2 \). Our task is to solve for \( x \).

Step 2: Use the Power Rule for Exponents

To eliminate the fractional exponent, raise both sides of the equation to the reciprocal of \( \frac{2}{5} \), which is \( \frac{5}{2} \), giving us \( \left(x^{\frac{2}{5}}\right)^{\frac{5}{2}} = 2^{\frac{5}{2}} \).

Step 3: Simplify Using the Property of Exponents

The left-hand side simplifies to \( x \) because \( \left(x^{\frac{2}{5}}\right)^{\frac{5}{2}} \) equals \( x^{1} \). So, the equation becomes \( x = 2^{\frac{5}{2}} \).

Step 4: Calculate the Right-hand Side

Calculate \( 2^{\frac{5}{2}} \). First, \( 2^{5} = 32 \), then take the square root, \( \sqrt{32} \), which equals \( 4\sqrt{2} \).

Step 5: Final Answer

Thus, \( x = 4\sqrt{2} \), which is approximately 5.656.

Key Concepts

Isolate the Variable

Isolating the variable is a crucial first step in solving equations, especially when dealing with exponents. The goal here is to have the term containing the variable on one side of the equation and everything else on the other. This allows us to focus on manipulating just one part of the equation to solve for the unknown variable. In this exercise, the expression with the variable, \( x^{\frac{2}{5}} \), is already alone on the left side of the equation: \( x^{\frac{2}{5}} = 2 \).
This means our job to isolate the variable is straightforward and already completed; we just need to understand how it facilitates a clear path to the next steps of solving the equation.
When variables are not initially isolated, you must perform operations to both sides of the equation to move other numbers or terms over to the opposite side.
For example, if you had \( 3x^{\frac{2}{5}} = 6\), you would divide both sides by 3 to isolate the variable expression, resulting in \( x^{\frac{2}{5}} = 2 \).

• Identify where the variable is located.
• Perform operations to keep the variable expression alone.

Power Rule for Exponents

The power rule for exponents is a valuable tool when working with equations that have fractional exponents. This rule states that \( (a^m)^n = a^{m \times n} \). In cases with fractional exponents, it helps us "undo" the exponent by raising both sides of the equation to the reciprocal of the fractional exponent.
For our exercise where \( x^{\frac{2}{5}} = 2 \), we use this rule by taking both sides of the equation to the power of \( \frac{5}{2} \): \[ \left( x^{\frac{2}{5}} \right)^{\frac{5}{2}} = 2^{\frac{5}{2}} \]
This action effectively cancels out the original fraction exponent, leaving us with \( x^1 \), or just \( x \), on the left.
The power rule simplifies these complex-looking equations into something more manageable, putting us in a position to solve for the variable.
This is incredibly useful not just for isolating variables but also for simplifying complex expressions in more extensive algebraic manipulations.

• Look for fractional exponents.
• Apply the power rule using the reciprocal to simplify.

Simplifying Exponents

After applying the power rule, it's time to simplify any remaining complicated exponents. This involves calculating expressions like \( 2^{\frac{5}{2}} \). Simplifying such expressions usually requires two steps: simplifying the base power and then applying the root.
First, calculate \( 2^5 \), which is 32. Then, "take the 2nd root," or the square root of 32, since 2 is in the denominator of the fraction exponent.
So, \[ \sqrt{32} = 4\sqrt{2} \]
This means our expression \( 2^{\frac{5}{2}} \) simplifies to \( 4\sqrt{2} \).
Simplifying means converting these awkward and unfamiliar mathematical notations into more user-friendly numbers and forms.

• Evaluate integer exponents first.
• Address the root of the simplified base.

By simplifying step-by-step, you maintain the equation's integrity and find the solution, making the math much easier to understand.