12. Use $i$ To Write $\sqrt{-49}$ As A Complex Number. $\square$13. Write The Expression As A Complex Number In Standard Form: $-5i(8-9i) = $ $\square$14. Use The Imaginary Number $i$ To Rewrite

What are Complex Numbers?

Complex numbers are mathematical expressions that combine real and imaginary numbers. They are used to represent quantities that have both magnitude and direction. In this article, we will explore how to simplify expressions involving complex numbers using the imaginary unit $i$.

Simplifying Expressions with Imaginary Numbers

12. Simplifying $\sqrt{-49}$ as a Complex Number

To simplify $\sqrt{-49}$ as a complex number, we can use the imaginary unit $i$. Recall that $i$ is defined as the square root of $-1$, i.e., $i = \sqrt{-1}$. We can rewrite $\sqrt{-49}$ as $\sqrt{49} \cdot \sqrt{-1}$.

\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i

Therefore, $\sqrt{-49}$ can be written as a complex number in standard form as $7i$.

13. Simplifying $-5i(8-9i)$ as a Complex Number

To simplify the expression $-5i(8-9i)$, we can use the distributive property to expand the product.

-5i(8-9i) = -5i(8) + (-5i)(-9i) = -40i + 45i^2

Recall that $i^2 = -1$, so we can substitute this value into the expression.

-40i + 45i^2 = -40i + 45(-1) = -40i - 45

Therefore, the expression $-5i(8-9i)$ can be written as a complex number in standard form as $-45 - 40i$.

14. Simplifying Expressions with Imaginary Numbers

To simplify expressions involving imaginary numbers, we can use the following properties:

• $i^2 = -1$
• $i^3 = i^2 \cdot i = -i$
• $i^4 = i^2 \cdot i^2 = 1$

We can use these properties to simplify expressions involving powers of $i$.

Example 1: Simplifying $i^5$

To simplify $i^5$, we can use the property $i^4 = 1$.

i^5 = i^4 \cdot i = 1 \cdot i = i

Therefore, $i^5 = i$.

Example 2: Simplifying $i^6$

To simplify $i^6$, we can use the property $i^4 = 1$.

i^6 = i^4 \cdot i^2 = 1 \cdot (-1) = -1

Therefore, $i^6 = -1$.

Example 3: Simplifying $i^7$

To simplify $i^7$, we can use the property $i^4 = 1$.

i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i

Therefore, $i^7 = -i$.

Example 4: Simplifying $i^8$

To simplify $i^8$, we can use the property $i^4 = 1$.

i^8 = i^4 \cdot i^4 = 1 \cdot 1 = 1

Therefore, $i^8 = 1$.

Conclusion

In this article, we have explored how to simplify expressions involving complex numbers using the imaginary unit $i$. We have used the properties of $i$ to simplify expressions involving powers of $i$ and have shown how to rewrite expressions in standard form. By understanding how to simplify expressions involving complex numbers, we can better work with these numbers in a variety of mathematical contexts.

Key Takeaways

• Complex numbers are mathematical expressions that combine real and imaginary numbers.
• The imaginary unit $i$ is defined as the square root of $-1$.
• We can use the properties of $i$ to simplify expressions involving powers of $i$.
• We can rewrite expressions in standard form using the properties of $i$.

Practice Problems

1. Simplify $\sqrt{-64}$ as a complex number.
2. Simplify $-3i(2+5i)$ as a complex number.
3. Simplify $i^9$ as a complex number.
4. Simplify $i^{11}$ as a complex number.

Answer Key

1. $\sqrt{-64} = 8i$
2. $-3i(2+5i) = -6i - 15i^2 = -6i + 15 = 15 - 6i$
3. $i^9 = i^8 \cdot i = 1 \cdot i = i$
4. $i^{11} = i^{10} \cdot i = (i^2)^5 \cdot i = (-1)^5 \cdot i = -i$
   Complex Numbers: Q&A =====================

Frequently Asked Questions

Q: What is a complex number?

A: A complex number is a mathematical expression that combines real and imaginary numbers. It is typically written in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is defined as the square root of $-1$. It is denoted by the letter $i$ and is used to represent the imaginary part of a complex number.

Q: How do I simplify a complex number?

A: To simplify a complex number, you can use the following steps:

1. Multiply the real and imaginary parts by the same value.
2. Combine like terms.
3. Simplify the expression using the properties of $i$.

Q: What are the properties of $i$?

A: The properties of $i$ are:

• $i^2 = -1$
• $i^3 = i^2 \cdot i = -i$
• $i^4 = i^2 \cdot i^2 = 1$

Q: How do I add complex numbers?

A: To add complex numbers, you can use the following steps:

1. Add the real parts.
2. Add the imaginary parts.
3. Combine like terms.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you can use the following steps:

1. Subtract the real parts.
2. Subtract the imaginary parts.
3. Combine like terms.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the following steps:

1. Multiply the real parts.
2. Multiply the imaginary parts.
3. Combine like terms.

Q: How do I divide complex numbers?

A: To divide complex numbers, you can use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator.
2. Simplify the expression using the properties of $i$.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of $a + bi$ is $a - bi$.

Q: How do I simplify expressions involving powers of $i$?

A: To simplify expressions involving powers of $i$, you can use the following properties:

• $i^2 = -1$
• $i^3 = i^2 \cdot i = -i$
• $i^4 = i^2 \cdot i^2 = 1$

Q: How do I simplify expressions involving complex numbers and real numbers?

A: To simplify expressions involving complex numbers and real numbers, you can use the following steps:

1. Multiply the real and imaginary parts by the same value.
2. Combine like terms.
3. Simplify the expression using the properties of $i$.

Q: What are some common mistakes to avoid when working with complex numbers?

A: Some common mistakes to avoid when working with complex numbers include:

• Forgetting to use the properties of $i$.
• Not simplifying expressions involving powers of $i$.
• Not combining like terms.

Conclusion

In this article, we have answered some frequently asked questions about complex numbers. We have covered topics such as simplifying complex numbers, adding and subtracting complex numbers, multiplying and dividing complex numbers, and simplifying expressions involving powers of $i$. By understanding these concepts, you can better work with complex numbers in a variety of mathematical contexts.

Key Takeaways

• Complex numbers are mathematical expressions that combine real and imaginary numbers.
• The imaginary unit $i$ is defined as the square root of $-1$.
• We can use the properties of $i$ to simplify expressions involving powers of $i$.
• We can rewrite expressions in standard form using the properties of $i$.

Practice Problems

1. Simplify $\sqrt{-64}$ as a complex number.
2. Simplify $-3i(2+5i)$ as a complex number.
3. Simplify $i^9$ as a complex number.
4. Simplify $i^{11}$ as a complex number.

Answer Key

1. $\sqrt{-64} = 8i$
2. $-3i(2+5i) = -6i - 15i^2 = -6i + 15 = 15 - 6i$
3. $i^9 = i^8 \cdot i = 1 \cdot i = i$
4. $i^{11} = i^{10} \cdot i = (i^2)^5 \cdot i = (-1)^5 \cdot i = -i$