%5E1%2F2.jpeg "(-81)^1/2")
Understanding (-81)^(1/2)
The expression (-81)^(1/2) represents the square root of -81. Understanding this requires knowledge of both real numbers and complex numbers.
Real Numbers and Square Roots
In the realm of real numbers, the square root of a number is defined as the number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. However, there's no real number that, when multiplied by itself, equals -81. This is because the square of any real number (positive or negative) is always positive.
Introducing Complex Numbers
To address this, we introduce complex numbers, which extend the real number system. Complex numbers involve the imaginary unit i, defined as the square root of -1 (i.e., i² = -1).
Calculating (-81)^(1/2)
Using complex numbers, we can calculate the square root of -81:
- Factor out -1: (-81)^(1/2) = (81 * -1)^(1/2)
- Apply the square root property: (81 * -1)^(1/2) = 81^(1/2) * (-1)^(1/2)
- Simplify: 81^(1/2) = 9 and (-1)^(1/2) = i
- Combine: 9 * i = 9i
Therefore, (-81)^(1/2) = 9i. This is a complex number with a real part of 0 and an imaginary part of 9.
Conclusion
While the square root of -81 doesn't exist within the real number system, it can be represented as a complex number using the imaginary unit i. Understanding complex numbers allows us to expand our mathematical understanding and solve problems that are not possible within the realm of real numbers alone.