Problem 65 - College Algebra with Modeling and Visualization

Question

Solve the equation. Check your answers. $$ x^{2 / 5}=4 $$

Step-by-Step Solution

Answer

The solution is \(x = 32\).

Step 1: Understand the Equation

The given equation is \( x^{2/5} = 4 \). This means we need to find the value of \( x \) such that raising it to the power of \( \frac{2}{5} \) results in 4.

Step 2: Isolate the Variable

To solve for \( x \), we rewrite the equation in terms of a power that can easily be solved. Raise both sides of the equation to the power of \( \frac{5}{2} \) to get rid of the fractional exponent: \((x^{2/5})^{5/2} = 4^{5/2} \).

Step 3: Simplify the Left Side

When you raise an exponent to another power, you multiply the exponents. Hence, \((x^{2/5})^{5/2} = x^{(2/5)*(5/2)} = x^1 = x\).

Step 4: Evaluate the Right Side

Calculate \(4^{5/2}\). First, find \(4^{1/2} = 2\), because the square root of 4 is 2. Then raise 2 to the 5th power: \[2^5 = 32\].

Step 5: Check the Solution

Substitute \(x = 32\) back into the original equation to verify the solution: \[32^{2/5} = (32^{1/5})^2 = 2^2 = 4\], which holds true.

Key Concepts

Fractional Exponents

Fractional exponents might seem intimidating at first, but they are quite manageable once you understand their meaning. A fractional exponent, such as \( x^{2/5} \), is another way of expressing a root and a power. The denominator of the fraction (in this case, 5) indicates the root to be taken; hence \( x^{2/5} \) is equivalent to the 5th root of \( x^2 \). The numerator is the power to which the base is raised after taking the root.

To solve equations with fractional exponents, you often have to manipulate them by getting rid of the fraction. This can be done by raising both sides of the equation to a reciprocal power of the fraction. In our example \( x^{2/5} = 4 \), we raised both sides to the power of \( \frac{5}{2} \), effectively eliminating the fraction and simplifying the left side directly to \( x \). This technique is extremely useful when dealing with equations that involve fractional exponents.

Equation Verification

Checking your solution is a crucial step when solving equations, as it confirms the correctness of your answer. Verification involves substituting your found solution back into the original equation to see if it makes a true statement. Consider our example where we solved for \( x \) and found \( x = 32 \).

To verify, substitute \( x = 32 \) back into the original equation.

• Calculate \( 32^{2/5} \): Start by determining the 5th root of 32, which is 2 because \( 2^5 = 32 \).
• Then, raise that result (2) to the power of 2: \( 2^2 = 4 \).

Since \( 32^{2/5} = 4 \), the solution is verified as correct. This comprehensive approach ensures you haven't made any errors along the way and strengthens your understanding of the problem-solving process.

Exponentiation

Exponentiation, the process of raising a number to a power, is a fundamental concept in mathematics. It allows us to express repeated multiplication of a number concisely. For real numbers, this concept applies equally to integers, fractions, and irrational numbers.

When working with exponents, a few rules significantly simplify calculations:

• The "power of a power" rule states that \((a^m)^n = a^{m\cdot n}\). This is particularly useful when dealing with fractional exponents.
• Base-exponential relationships help evaluate roots using powers, such as finding that \(4^{1/2} = 2\) because the square root of 4 is 2.
• Multiplying exponents means adding their values, e.g., \(a^m \cdot a^n = a^{m+n}\).

For our problem, understanding these rules helped evaluate the power and find \(4^{5/2}\) swiftly, splitting it into manageable steps: finding much simpler powers and roots first. Mastering these concepts unlocks the ability to tackle more complex exponential equations confidently.