What is the square root of negative 81?
Understand the Problem
The question is asking for the square root of a negative number, specifically -81. Since negative numbers do not have real square roots, this question introduces the concept of complex numbers, where the square root of -1 is denoted as 'i'. Therefore, the square root of -81 can be expressed as 9i.
Answer
$9i$
Answer for screen readers
The square root of -81 is $9i$.
Steps to Solve
- Identify the square root of the negative number
To find the square root of -81, we first recognize that we can factor out the negative sign. This is done by rewriting -81 as $-1 \cdot 81$.
- Use the definition of complex numbers
We know that the square root of -1 is defined as $i$. Thus, we can express the square root of -81 as:
$$ \sqrt{-81} = \sqrt{-1 \cdot 81} $$
- Separate the square root components
Using the property of square roots, we can separate the components:
$$ \sqrt{-1 \cdot 81} = \sqrt{-1} \cdot \sqrt{81} $$
- Calculate the square roots
Now we can substitute the known values:
- $\sqrt{-1} = i$
- $\sqrt{81} = 9$
Thus,
$$ \sqrt{-81} = i \cdot 9 $$
- Combine the results
Finally, we combine the results to express the square root of -81 in its simplest form:
$$ \sqrt{-81} = 9i $$
The square root of -81 is $9i$.
More Information
In mathematics, any square root of a negative number involves the imaginary unit, represented as $i$. The concept of imaginary numbers is essential in advanced mathematics, particularly in fields like engineering and physics.
Tips
- Not recognizing the imaginary unit: Some may attempt to take the square root of a negative number directly without using the imaginary unit, which leads to incorrect conclusions. Always remember to express it in terms of $i$.
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