%5E1%2F4.jpeg "(-81)^1/4")
Understanding (-81)^(1/4)
The expression (-81)^(1/4) represents the fourth root of -81. Let's break down what this means and how to approach it.
Roots and Exponents
- Roots are the opposite of exponents. If a number raised to a power equals another number, then the root of the second number is the original number.
- The fourth root of a number is the number that, when multiplied by itself four times, equals the original number. For example, the fourth root of 81 is 3 because 3 * 3 * 3 * 3 = 81.
Dealing with Negatives
- Even roots (like the fourth root) of negative numbers are not real numbers. This is because any real number multiplied by itself an even number of times results in a positive number.
- Imaginary Numbers: To handle this, we introduce the imaginary unit, denoted by i, where i² = -1.
Solving (-81)^(1/4)
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Real Number Solution: There is no real number solution to (-81)^(1/4) because no real number multiplied by itself four times can equal a negative number.
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Complex Number Solution: We can express the solution using complex numbers:
- The fourth root of -81 can be written as (-1)^(1/4) * (81)^(1/4).
- (81)^(1/4) is simply 3 (because 3 * 3 * 3 * 3 = 81).
- (-1)^(1/4) has four solutions in the complex plane.
Therefore, the four solutions for (-81)^(1/4) are:
- 3 * (1 + i) / √2
- 3 * (-1 + i) / √2
- 3 * (-1 - i) / √2
- 3 * (1 - i) / √2
In summary, while there's no real number solution to (-81)^(1/4), it has four solutions in the realm of complex numbers.