Problem 62 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{5}+32=0\)

Step-by-Step Solution

Answer

The real solution is \(x = -2\).

Step 1: Isolate the Power Term

The given equation is \(x^5 + 32 = 0\). First, we need to isolate the power term. Subtract 32 from both sides to get \(x^5 = -32\).

Step 2: Apply the Fifth Root

To solve for \(x\), we need to take the fifth root of both sides of the equation. Thus, we calculate the fifth root of \(-32\) to get \(x = \sqrt[5]{-32}\).

Step 3: Simplify the Root

The fifth root of \(-32\) is \(-2\), since \((-2)^5 = -32\). Therefore, \(x = -2\).

Step 4: Verify the Solution

Verify the solution by substituting \(x = -2\) back into the original equation: \((-2)^5 + 32 = -32 + 32 = 0\). This confirms that \(x = -2\) is indeed a solution.

Key Concepts

Real Solutions

In mathematics, finding the real solutions of an equation involves identifying the values that a variable can take which make the equation true. Real solutions are all the solutions that are represented by real numbers, excluding any complex numbers.
For the equation we are solving, which is \(x^5 + 32 = 0\), our goal is to find all values of \(x\) that satisfy this equation in the realm of real numbers. Real numbers include both positive numbers, negative numbers, and zero.

To begin, we move terms to isolate the variable with the power: \(x^5 = -32\). This isolates the power term, making it easier to solve for \(x\). It is important to recognize that not all equations involving powers have real solutions, but in this particular case, we will prove that a real solution exists by simplifying and solving \(x^5 = -32\) to find a real \(x\) value.

Fifth Root

Taking the root of a number is an inverse operation to raising a number to a power. For this particular problem, we need to find the fifth root of a number.

To solve \(x^5 = -32\), we find \(x\) by computing the fifth root of \(-32\). The process of taking a fifth root can be symbolized as \(x = \sqrt[5]{-32}\). This operation is asking us what number, when multiplied by itself five times, results in \(-32\).

• Finding an odd root (like the fifth root) of a negative number is possible and results in a negative real answer.
• Here, this means our real solution will be some negative number.

After calculating \(\sqrt[5]{-32}\), we find that it equals \(-2\) because \((-2)^5 = -32\). Finding roots requires a good understanding of powers and decomposition of numbers, which involves practicing breaking down an equation into simpler steps or calculations.

Isolating Power Terms

When faced with an equation that involves power terms, like \(x^5\) in our exercise, it's crucial to isolate these terms first to simplify solving the equation.

To isolate the term \(x^5\) in the equation \(x^5 + 32 = 0\), we subtract 32 from both sides. This gives us \(x^5 = -32\).

• Isolating terms helps focus on resolving the part of the equation that contains power terms.
• It sets up the problem so we can apply a root, which simplifies the equation further.
• Isolating power terms is typically the first major step before applying roots or other operations.

By transforming the equation to \(x^5 = -32\), we prepare to take the appropriate root. In this equation, simplifying further with the fifth root reveals the real solution, \(x = -2\), verifying the calculation by plugging it back shows the equation holds true at \(x = -2\). Learning to isolate terms skillfully makes solving equations more straightforward and approachable.