Problem 21 - A Graphical Approach to College Algebra

Question

Solve each equation by hand. Do not use a calculator. $$x^{2 / 5}=4$$

Step-by-Step Solution

Answer

The solution is \( x = 32 \).

Step 1: Isolate the term with the variable

In the given equation, the term with the variable is already isolated: \[ x^{ \frac{2}{5} } = 4 \] There is no other simplification needed in this step.

Step 2: Eliminate the fractional exponent

To solve the equation, we need to eliminate the fractional exponent. Raise both sides of the equation to the power of \( \frac{5}{2} \), which is the reciprocal of \( \frac{2}{5} \).\[ \left( x^{ \frac{2}{5} } \right)^{ \frac{5}{2} } = 4^{ \frac{5}{2} } \]The left side simplifies to \( x \), while the right side remains to be calculated:

Step 3: Calculate the right side

Compute \( 4^{ \frac{5}{2} } \):1. First, calculate \( 4^{1/2} \), which is the square root of 4: \( 2 \).2. Then, raise the result to the 5th power: \( 2^5 = 32 \).Now we know that \( 4^{ \frac{5}{2} } = 32 \).

Step 4: Write the final solution

After solving the equation, we find that the solution is:\[ x = 32 \]This completes the process of solving the equation.

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, represent expressions that involve roots and powers. The numerator of a fractional exponent indicates the power, while the denominator shows the root. For example, in the expression \( x^{\frac{2}{5}} \), the number 2 indicates that the value is squared, and the denominator 5 represents the fifth root.

When working with fractional exponents, it’s often helpful to break the process into two steps:

• First, take the root as indicated by the denominator.
• Then, raise the result to the power suggested by the numerator.

Using fractional exponents allows for an easier manipulation of expressions, particularly when it is necessary to reverse operations, such as eliminating exponents in an equation. This makes solving equations with fractional exponents a much more straightforward task, once you have isolated the term correctly.

Equation Isolation

Equation isolation involves rearranging the given equation to make the variable of interest the subject of the equation. This means that the variable is on one side of the equal sign, separated from other components of the equation.

In the original problem \( x^{\frac{2}{5}} = 4 \), the term \( x^{\frac{2}{5}} \) is already isolated. This means there's nothing more to solve or rearrange in terms of variable isolation in this instance.

It's always essential to isolate the term with the variable when solving equations, because that paves the way for applying further mathematical operations straightforwardly. Isolation simplifies complex expressions, allowing you to focus on the variable itself and the operations needed to solve the equation.

Exponentiation

Exponentiation refers to the mathematical operation involving exponents, which describes how many times a number, referred to as the base, is multiplied by itself. For example, \( 4^2 = 16 \) indicates that 4 is multiplied by itself once.

In relation to the exercise, once the equation is isolated, we use a method involving exponentiation to remove the fractional exponent. The reciprocal of the original exponent, \( \frac{5}{2} \), is used to elevate both sides of the equation.

This results in:

• \((x^{\frac{2}{5}})^{\frac{5}{2}} = 4^{\frac{5}{2}}\)
• The left side simplifies to \( x \), since raising a power to its reciprocal results in the base itself.

Next, calculate the right side: Take the square root of 4 to get 2, then raise 2 to the 5th power, resulting in 32.

Thus, the solution is \( x = 32 \). Exponentiation provides a powerful mechanism for resolving equations that contain exponents by reversing the original operations through the use of reciprocal powers.