Problem 51 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation. $$x^{4}=256$$

Step-by-Step Solution

Answer

The solutions are \( x = 4 \) and \( x = -4 \).

Step 1: Rewrite the Equation

The equation given is \( x^4 = 256 \). Our first task is to rewrite this equation in a form that helps us find the solutions. Note that 256 is a power of 2: \( 256 = 2^{8} \). Therefore, the equation can be rewritten as \( x^4 = (2^{4})^2 \), which means \( x^4 = (2^2)^4 \).

Step 2: Equate the Powers

Since \( x^4 = (2^2)^4 \), comparing the bases, we can equate \( x^4 = (4)^4 \). Therefore, the expression simplifies to \( x = 4 \) or \( x = -4 \) since taking the fourth root gives us both a positive and negative root.

Step 3: Verify Solutions

To ensure that both values satisfy the original equation \( x^4 = 256 \), substitute back: When \( x = 4 \), \( 4^4 = 256 \). When \( x = -4 \), \( (-4)^4 = 256 \). Thus, both solutions are verified.

Key Concepts

Fourth Roots

Taking the fourth root of a number is a process that allows you to determine which numbers multiplied by themselves four times result in the target number. Just as a square root asks "what number, when squared, produces this value?"; similarly, a fourth root asks, "what number when raised to the power of four equals this value?" For example, in the original exercise involving the equation \(x^4 = 256\), finding the fourth roots of 256 is crucial because they are the solutions to the equation.

When calculating fourth roots, it's important to consider both positive and negative possibilities. A number raised to an even power, such as four, will result in a positive number, even if the original number was negative. That's why both 4 and -4 are considered solutions since both \(4^4\) and \((-4)^4\) result in 256.

To summarize:

• The fourth root of a positive number results in both a positive and negative number.
• Even powers negate any negative signs, resulting in a positive product.
• Finding roots involves reverse operations on powers.

Exponential Equation

An exponential equation is an equation in which variables appear as exponents. These kinds of equations are pervasive in mathematical models representing growth processes, decay, and other real-world phenomena.

In our given equation, \(x^4 = 256\), the variable \(x\) is in the exponent position. This indicates that the solution involves understanding powers and roots. Rewriting equations in exponential form often aids in solving them, as seen when we described 256 as \((2^2)^4\).

Breaking it down further:

• The equation was simplified from \(x^4 = 256\) to \(x^4 = (2^2)^4\).
• This manipulation involves recognizing 256 as a power of 2, specifically \(2^8\), and further breaking it down into a suitable form for comparison.
• It allowed us to solve the equation by equating the bases, simplifying the equation to \(x = \pm 4\).

Recognizing how to transform an exponential equation is key. It often involves creative rewritings like factoring or assessing a known base representation, which simplifies comparison and solution finding.

Solution Verification

Verifying your solution is one of the most essential steps in solving any equation. It confirms whether your solution is accurate and fits the equation as intended. It involves substituting your solution back into the original equation to ensure everything balances.

In our example, we found solutions \(x = 4\) and \(x = -4\). Verification means checking that, when substituted back into the original equation, both solutions satisfy \(x^4 = 256\).

• For \(x = 4\), substitution gives \(4^4 = 256\), confirming the first solution.
• For \(x = -4\), substitution yields \((-4)^4 = 256\), confirming the second solution.

It's important because:

• It confirms the integrity and correctness of your solution.
• Measurement against the initial equation ensures no errors in reasoning or calculation.
• Verification is the final checkpoint before concluding an analysis, making the process foolproof.