Find the Complex Conjugate -8+ square root of -25
This problem asks for the complex conjugate of a given complex number. A complex number is composed of a real part and an imaginary part, often expressed as a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (the square root of -1). The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. So the task is to determine the complex conjugate of the complex number -8 plus the square root of -25, which would involve changing the sign of its imaginary component.
$- 8 + \sqrt{- 25}$
Answer
Solution:
Step 1: Simplify the expression.
- Start by breaking down the complex number into its components.
Step 1.1: Express the negative number inside the square root.
- Rewrite $-25$ as $-1 \times 25$. Thus, the expression becomes $-8 + \sqrt{-1 \times 25}$.
Step 1.2: Separate the square root of the product into the product of square roots.
- Convert $\sqrt{-1 \times 25}$ into $\sqrt{-1} \cdot \sqrt{25}$. Now the expression is $-8 + \sqrt{-1} \cdot \sqrt{25}$.
Step 1.3: Identify the imaginary unit.
- Recognize that $\sqrt{-1}$ is the imaginary unit $i$. The expression now reads $-8 + i \cdot \sqrt{25}$.
Step 1.4: Simplify the square root of a perfect square.
- Notice that $25$ is a perfect square, $5^2$. Rewrite the expression as $-8 + i \cdot \sqrt{5^2}$.
Step 1.5: Extract the square root of the perfect square.
- Since the square root of $5^2$ is $5$, the expression simplifies to $-8 + i \cdot 5$.
Step 1.6: Rearrange the terms.
- Place the real number before the imaginary unit, resulting in $-8 + 5i$.
Step 2: Determine the complex conjugate.
- To find the complex conjugate, change the sign of the imaginary part. The complex conjugate is $-8 - 5i$.
Knowledge Notes:
The complex conjugate of a complex number is found by changing the sign of the imaginary part. In general, if a complex number is written as $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$), its complex conjugate is $a - bi$.
Here are the steps we followed to find the complex conjugate:
Simplify the Complex Number: Break down the complex number into its real and imaginary parts. If the number includes a square root of a negative number, express it in terms of the imaginary unit $i$.
Separate Square Roots: If the square root of a negative number is encountered, separate it into the product of the square root of $-1$ (which is $i$) and the square root of the positive component.
Identify Perfect Squares: Simplify the square roots of perfect squares to their integer values.
Rearrange Terms: Write the complex number in standard form, $a + bi$.
Find the Complex Conjugate: Change the sign of the imaginary part to find the complex conjugate, resulting in $a - bi$.
Understanding these steps is crucial for working with complex numbers, as complex conjugates play a significant role in various mathematical operations, including simplifying the division of complex numbers and finding polynomial roots.
