Question

Find the real solution(s) of the equation. Round your answer to two decimal places when appropriate. (See Example 4.) \(\frac{1}{3} x^4=27\)

Short answer

The equation \(\frac{1}{3} x^4=27\) has two real solutions, \( x = 3 \) and \( x = -3 \).

Step-by-step solution

Simplify the Equation

First, clear fraction by multiplying both sides of the equation by 3 to get \( x^4 = 81 \). In this way, we have simplified our original equation by isolating \(x^4\) on one side of the equation.

Take the Fourth Root

Since \( x^4 = 81 \), then \( x \) would be any number such that raising that number to the power of 4 will result in 81. In this case, we need to take the fourth root of both sides of the equation to solve for \(x\). Now, \( x = \pm \sqrt[4]{81} \). This operation gives us the positive or negative root of 81.

Calculate the Value

Now, by calculating the value of the fourth root of 81, we find that the real solutions of the equation are \( x = \pm 3 \).

Key concepts

Understanding Fourth Roots

In the context of solving polynomial equations, the term 'fourth root' appears frequently when we encounter a variable raised to the fourth power, like in the given exercise. Imagine you have a number, and when you multiply it by itself three more times (for a total of four instances), you reach a specific value. To reverse this process - to start from this specific value and find the original number - you would extract the fourth root.

For example, if we have the equation \( x^4 = 81 \), the fourth root will undo the exponent of 4. Mathematically, it is denoted as \( \sqrt[4]{81} \), and it simplifies to find the value of \(x\) that satisfies the original equation when raised to the fourth power. The fourth root of 81 is 3 because \(3^4 = 3 × 3 × 3 × 3 = 81\). However, it's essential to consider that positive numbers have two real fourth roots, a positive and a negative one, since \( (-3)^4 = 81 \) as well. Therefore, the fourth roots of 81 are \(±3\).

Isolating Variables

When solving equations, 'isolating the variable' means rearranging the equation so that the variable we want to solve for stands alone on one side of the equation. This is a key step in the solution process as it simplifies the complexity of the equation and provides a clearer path to finding the solution.

In our given exercise, we start with \(\frac{1}{3} x^4 = 27\) and want to isolate \(x^4\). We accomplish this by multiplying both sides of the equation by 3, removing the fraction, and thus isolating \(x^4\) on one side: \(x^4 = 81\). This step moves us closer towards finding the value of \(x\) by setting the stage for extracting the fourth root without additional constants or coefficients complicating the operation.

Simplifying Equations

The process of 'simplifying equations' is the act of converting an equation into the simplest form possible, making it easier to solve. This involves performing operations to both sides of the equation until the variable is isolated and the equation appears less cluttered. Simplification may include expanding expressions, clearing fractions, combining like terms, and reducing expressions to least terms among other operations.

In the exercise, after isolating \(x^4\) by multiplying by 3, the equation becomes much simpler and easier to solve. The simplification here is also partly due to knowing that the fourth root of 81 is a much more straightforward operation to perform compared to finding the fourth root of a fraction. Simplified equations often lead directly to finding the solution more efficiently and with fewer opportunities for errors.