Find the Complex Conjugate -8+ square root of -25

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:

1. 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 $i$.

2. 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.

3. Identify Perfect Squares: Simplify the square roots of perfect squares to their integer values.

4. Rearrange Terms: Write the complex number in standard form, $a+bi$.

5. 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.