Simplify. Show All Work.12. $\sqrt{-24}$13. ( 5 + 7 I ) − ( − 2 + 10 I (5+7i) - (-2+10i ( 5 + 7 I ) − ( − 2 + 10 I ]

12. Simplifying Square Roots of Negative Numbers

When simplifying square roots of negative numbers, we need to express the number in terms of its prime factors. In this case, we have $\sqrt{-24}$. To simplify this expression, we can start by factoring the number $-24$ into its prime factors.

Factoring -24

$-24 = -1 \times 24$

$24 = 2 \times 2 \times 2 \times 3$

So, we can write $-24$ as:

$-24 = -1 \times 2 \times 2 \times 2 \times 3$

Now, we can take the square root of each factor:

$\sqrt{-24} = \sqrt{-1} \times \sqrt{2 \times 2 \times 2 \times 3}$

$\sqrt{-24} = i \times \sqrt{2 \times 2 \times 2 \times 3}$

$\sqrt{-24} = i \times 2 \times 2 \times \sqrt{3}$

$\sqrt{-24} = 4i\sqrt{3}$

Therefore, the simplified form of $\sqrt{-24}$ is $4i\sqrt{3}$.

13. Simplifying Complex Numbers

When simplifying complex numbers, we need to combine like terms. In this case, we have $(5+7i) - (-2+10i)$. To simplify this expression, we can start by distributing the negative sign to the terms inside the parentheses.

Distributing the Negative Sign

$(5+7i) - (-2+10i)$

$= 5 + 7i + 2 - 10i$

Now, we can combine like terms:

$= 5 + 2 + 7i - 10i$

$= 7 + (-3)i$

Therefore, the simplified form of $(5+7i) - (-2+10i)$ is $7 - 3i$.

Simplifying Complex Numbers: A Step-by-Step Guide

Simplifying complex numbers can be a challenging task, but with a step-by-step approach, it can be made easier. Here's a guide on how to simplify complex numbers:

Step 1: Identify the Complex Number

The first step in simplifying a complex number is to identify the complex number. This involves recognizing the real and imaginary parts of the number.

Step 2: Distribute the Negative Sign (if necessary)

If the complex number is in the form of $(a+bi) - (c+di)$, we need to distribute the negative sign to the terms inside the parentheses.

Step 3: Combine Like Terms

Once we have distributed the negative sign, we can combine like terms. This involves adding or subtracting the real and imaginary parts separately.

Step 4: Simplify the Expression

After combining like terms, we can simplify the expression by writing it in the form of $a+bi$.

Tips for Simplifying Complex Numbers

Simplifying complex numbers can be a challenging task, but with some tips and tricks, it can be made easier. Here are some tips for simplifying complex numbers:

Tip 1: Use the Distributive Property

When simplifying complex numbers, it's often helpful to use the distributive property. This involves distributing the negative sign to the terms inside the parentheses.

Tip 2: Combine Like Terms

When combining like terms, it's essential to add or subtract the real and imaginary parts separately.

Tip 3: Simplify the Expression

After combining like terms, we can simplify the expression by writing it in the form of $a+bi$.

Common Mistakes to Avoid When Simplifying Complex Numbers

When simplifying complex numbers, there are several common mistakes to avoid. Here are some common mistakes to avoid:

Mistake 1: Not Distributing the Negative Sign

When simplifying complex numbers, it's essential to distribute the negative sign to the terms inside the parentheses.

Mistake 2: Not Combining Like Terms

When combining like terms, it's essential to add or subtract the real and imaginary parts separately.

Mistake 3: Not Simplifying the Expression

After combining like terms, we need to simplify the expression by writing it in the form of $a+bi$.

Conclusion

Simplifying complex numbers can be a challenging task, but with a step-by-step approach, it can be made easier. By following the steps outlined in this guide, you can simplify complex numbers with ease. Remember to use the distributive property, combine like terms, and simplify the expression to get the correct answer. With practice and patience, you can become proficient in simplifying complex numbers.

Frequently Asked Questions About Simplifying Complex Numbers

Simplifying complex numbers can be a challenging task, but with the right guidance, it can be made easier. Here are some frequently asked questions about simplifying complex numbers, along with their answers.

Q1: What is a complex number?

A complex number is a number that can be expressed in the form of $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q2: How do I simplify a complex number?

To simplify a complex number, you need to follow these steps:

1. Distribute the negative sign (if necessary)
2. Combine like terms
3. Simplify the expression

Q3: What is the distributive property?

The distributive property is a mathematical property that allows you to distribute a negative sign to the terms inside the parentheses.

Q4: How do I combine like terms?

To combine like terms, you need to add or subtract the real and imaginary parts separately.

Q5: What is the imaginary unit?

The imaginary unit is a mathematical constant that is defined as the square root of -1. It is denoted by the letter $i$.

Q6: How do I simplify a complex number with a negative sign?

To simplify a complex number with a negative sign, you need to distribute the negative sign to the terms inside the parentheses and then combine like terms.

Q7: Can I simplify a complex number with a variable?

Yes, you can simplify a complex number with a variable. However, you need to follow the same steps as before: distribute the negative sign, combine like terms, and simplify the expression.

Q8: How do I simplify a complex number with a fraction?

To simplify a complex number with a fraction, you need to follow the same steps as before: distribute the negative sign, combine like terms, and simplify the expression.

Q9: Can I simplify a complex number with a decimal?

Yes, you can simplify a complex number with a decimal. However, you need to follow the same steps as before: distribute the negative sign, combine like terms, and simplify the expression.

Q10: How do I check my work when simplifying complex numbers?

To check your work when simplifying complex numbers, you need to follow these steps:

1. Plug in the values into the original expression
2. Simplify the expression
3. Check if the simplified expression is equal to the original expression

Common Mistakes to Avoid When Simplifying Complex Numbers

When simplifying complex numbers, there are several common mistakes to avoid. Here are some common mistakes to avoid:

Mistake 1: Not Distributing the Negative Sign

When simplifying complex numbers, it's essential to distribute the negative sign to the terms inside the parentheses.

Mistake 2: Not Combining Like Terms

When combining like terms, it's essential to add or subtract the real and imaginary parts separately.

Mistake 3: Not Simplifying the Expression

After combining like terms, we need to simplify the expression by writing it in the form of $a+bi$.

Tips for Simplifying Complex Numbers

Simplifying complex numbers can be a challenging task, but with some tips and tricks, it can be made easier. Here are some tips for simplifying complex numbers:

Tip 1: Use the Distributive Property

When simplifying complex numbers, it's often helpful to use the distributive property. This involves distributing the negative sign to the terms inside the parentheses.

Tip 2: Combine Like Terms

When combining like terms, it's essential to add or subtract the real and imaginary parts separately.

Tip 3: Simplify the Expression

After combining like terms, we can simplify the expression by writing it in the form of $a+bi$.

Conclusion

Simplifying complex numbers can be a challenging task, but with the right guidance, it can be made easier. By following the steps outlined in this guide, you can simplify complex numbers with ease. Remember to use the distributive property, combine like terms, and simplify the expression to get the correct answer. With practice and patience, you can become proficient in simplifying complex numbers.