Perform The Indicated Operation, And Write The Expression In The Standard Form \[$ A + Bi \$\].$\[ I^{12} + I^{10} - I^4 - I^2 \\]$\[ I^{12} + I^{10} - I^4 - I^2 = \square \\](Simplify Your Answer.)

Introduction

Complex numbers are an essential part of mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will focus on simplifying complex expressions, specifically the expression ${ i^{12} + i^{10} - i^4 - i^2 }$. We will use the standard form of a complex number, which is $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Understanding Complex Numbers

Before we dive into simplifying the given expression, let's briefly review the basics of complex numbers. A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. The imaginary unit $i$ is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$.

Properties of Complex Numbers

Complex numbers have several properties that are essential to understand when working with them. Some of the key properties include:

• Addition: The sum of two complex numbers is another complex number. For example, $(a + bi) + (c + di) = (a + c) + (b + d)i$.
• Multiplication: The product of two complex numbers is another complex number. For example, $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.
• Conjugate: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of $a + bi$ is $a - bi$.

Simplifying the Expression

Now that we have reviewed the basics of complex numbers, let's focus on simplifying the given expression ${ i^{12} + i^{10} - i^4 - i^2 }$. To simplify this expression, we need to use the properties of complex numbers, specifically the properties of exponents.

Using Exponent Rules

One of the key rules of exponents is that $i^4 = 1$. This is because $i^2 = -1$, and therefore $i^4 = (i^2)^2 = (-1)^2 = 1$. We can use this rule to simplify the expression.

Step 1: Simplify $i^{12}$

Using the rule $i^4 = 1$, we can simplify $i^{12}$ as follows:

$i^{12} = (i^4)^3 = 1^3 = 1$

Step 2: Simplify $i^{10}$

Using the rule $i^4 = 1$, we can simplify $i^{10}$ as follows:

$i^{10} = (i^4)^2 \cdot i^2 = 1^2 \cdot (-1) = -1$

Step 3: Simplify $i^4$

Using the rule $i^4 = 1$, we can simplify $i^4$ as follows:

$i^4 = 1$

Step 4: Simplify $i^2$

Using the rule $i^2 = -1$, we can simplify $i^2$ as follows:

$i^2 = -1$

Step 5: Combine the Terms

Now that we have simplified each term, we can combine them to get the final result:

$i^{12} + i^{10} - i^4 - i^2 = 1 + (-1) - 1 - (-1) = 0$

Conclusion

In this article, we have simplified the complex expression ${ i^{12} + i^{10} - i^4 - i^2 }$. We used the properties of complex numbers, specifically the properties of exponents, to simplify the expression. The final result is $0$, which is a real number.

Final Answer

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Introduction

In our previous article, we simplified the complex expression ${ i^{12} + i^{10} - i^4 - i^2 }$. We used the properties of complex numbers, specifically the properties of exponents, to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying complex expressions.

Q&A

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed without any imaginary part, whereas a complex number is a number that has both real and imaginary parts.

Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$.

Q: How do you simplify complex expressions?

A: To simplify complex expressions, you need to use the properties of complex numbers, specifically the properties of exponents. You can also use the rules of arithmetic operations, such as addition and multiplication.

Q: What is the rule for multiplying complex numbers?

A: The rule for multiplying complex numbers is $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.

Q: How do you find 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: What is the difference between a complex number and a polynomial?

A: A complex number is a number that can be expressed in the form $a + bi$, whereas a polynomial is an expression that consists of variables and coefficients.

Q: How do you simplify complex polynomials?

A: To simplify complex polynomials, you need to use the properties of complex numbers, specifically the properties of exponents. You can also use the rules of arithmetic operations, such as addition and multiplication.

Q: What is the rule for dividing complex numbers?

A: The rule for dividing complex numbers is $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$.

Q: How do you simplify complex fractions?

A: To simplify complex fractions, you need to use the properties of complex numbers, specifically the properties of exponents. You can also use the rules of arithmetic operations, such as addition and multiplication.

Q: What is the difference between a complex number and a matrix?

A: A complex number is a number that can be expressed in the form $a + bi$, whereas a matrix is a rectangular array of numbers.

Q: How do you simplify complex matrices?

A: To simplify complex matrices, you need to use the properties of complex numbers, specifically the properties of exponents. You can also use the rules of arithmetic operations, such as addition and multiplication.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying complex expressions. We have covered topics such as the difference between real and complex numbers, the imaginary unit $i$, and the rules for multiplying and dividing complex numbers. We hope that this article has been helpful in clarifying any doubts you may have had about simplifying complex expressions.

Final Answer

The final answer is $\boxed{0}$.