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Chapter 8: Problem 74

Solve. $$x^{4}+81=0$$

Short Answer

Expert verified
The solutions are \( x = 3i, -3i, 3i, -3i \).

Step by step solution

01

Move Constant Term

Start by isolating the term with the variable. Subtract 81 from both sides of the equation to get: \[ x^4 = -81 \]
02

Take the Fourth Root

To solve for \(x\), take the fourth root of both sides of the equation. Remember that taking the even root of a negative number introduces imaginary numbers: \[ x = \text{fourth root of } -81 \] So, we get: \[ x = \frac{\text{fourth root of } 81}{i} \]
03

Simplify Roots

Now, simplify the fourth root of 81 and consider the imaginary unit \(i\): \[ \text{Fourth root of } 81 = \text{(Fourth root of } 3^4) = 3 \] We have: \[ x = 3i \] and the negative versions: \[ x = -3i \]
04

Consider All Fourth Roots

Since it's the fourth root, there are multiple solutions, including complex numbers. These solutions are: \[ x = 3i, -3i, 3i, -3i \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fourth Root
The fourth root is a mathematical operation that asks which number, when multiplied by itself four times, gives the original number. For example, the fourth root of 81 is the number that, when raised to the fourth power, equals 81. This can be written as \(\text{fourth root of } 81 = 3\), since \(3^4 = 81\). When dealing with negative numbers, the concept becomes a bit trickier and introduces complex numbers. Taking the fourth root of a negative number involves imaginary numbers because no real number raised to an even power can result in a negative number.
Imaginary Numbers
Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit \(i\), where \(i\) is defined as \(\text{the square root of } -1\). Imaginary numbers are essential when dealing with even roots of negative numbers. In our exercise, when we take the fourth root of -81, we step into the realm of imaginary numbers. Simplifying the fourth root of -81, we get \(3i\), because \(3^4 = 81\) and we include \(i\) to indicate that it is the root of a negative number. This results in complex solutions like \(3i\) and \(-3i\).
Higher Degree Polynomials
Higher degree polynomials are equations that involve a variable raised to a power of three or higher. For example, a fourth-degree polynomial (like \(x^4\)) is one where the highest exponent of the variable is four. Solving these can become complex and often requires special techniques or factoring. The solutions to these polynomials can include real numbers, imaginary numbers, or a combination of both. Typically, a fourth-degree polynomial will have four solutions (or roots) which can be arranged into pairs of complex conjugates. In our exercise, the fourth-degree polynomial \(x^4 + 81 = 0\) has four solutions given as \(3i, -3i, 3i, -3i\).

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