Simplify The Expression: 5 I ⋅ ( − I 5i \cdot (-i 5 I ⋅ ( − I ]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. In this article, we will focus on simplifying the expression 5i⋅(−i)5i \cdot (-i), where ii is the imaginary unit. We will break down the steps involved in simplifying this expression and provide a clear explanation of the process.

Understanding the Imaginary Unit

Before we dive into simplifying the expression, let's briefly review the concept of the imaginary unit. The imaginary unit, denoted by ii, is a mathematical concept that is used to extend the real number system to the complex number system. It is defined as the square root of −1-1, i.e., i=−1i = \sqrt{-1}. The imaginary unit has a number of important properties, including:

  • i2=−1i^2 = -1
  • i3=−ii^3 = -i
  • i4=1i^4 = 1

Simplifying the Expression

Now that we have a basic understanding of the imaginary unit, let's focus on simplifying the expression 5i⋅(−i)5i \cdot (-i). To simplify this expression, we can use the distributive property of multiplication over addition. This property states that for any real numbers aa, bb, and cc, we have:

a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot c

Using this property, we can rewrite the expression 5i⋅(−i)5i \cdot (-i) as:

5i⋅(−i)=5i⋅(−1)⋅i5i \cdot (-i) = 5i \cdot (-1) \cdot i

Now, we can simplify the expression further by using the property i2=−1i^2 = -1. This gives us:

5i⋅(−1)⋅i=5i⋅(−i)=−5i25i \cdot (-1) \cdot i = 5i \cdot (-i) = -5i^2

Finally, we can simplify the expression −5i2-5i^2 by using the property i2=−1i^2 = -1. This gives us:

−5i2=−5(−1)=5-5i^2 = -5(-1) = 5

Conclusion

In this article, we have simplified the expression 5i⋅(−i)5i \cdot (-i) using the distributive property of multiplication over addition and the properties of the imaginary unit. We have shown that the expression simplifies to 55, which is a real number. This example illustrates the importance of simplifying expressions in mathematics and provides a clear understanding of the underlying concepts.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

  1. Rewrite the expression 5i⋅(−i)5i \cdot (-i) using the distributive property of multiplication over addition.
  2. Simplify the expression 5i⋅(−1)⋅i5i \cdot (-1) \cdot i by using the property i2=−1i^2 = -1.
  3. Simplify the expression −5i2-5i^2 by using the property i2=−1i^2 = -1.

Example Problems

Here are a few example problems that involve simplifying expressions with the imaginary unit:

  • Simplify the expression 3i⋅(2i)3i \cdot (2i).
  • Simplify the expression 4i⋅(−3i)4i \cdot (-3i).
  • Simplify the expression 2i⋅(5i)2i \cdot (5i).

Practice Problems

Here are a few practice problems that involve simplifying expressions with the imaginary unit:

  • Simplify the expression 6i⋅(−2i)6i \cdot (-2i).
  • Simplify the expression 9i⋅(3i)9i \cdot (3i).
  • Simplify the expression 8i⋅(−4i)8i \cdot (-4i).

Common Mistakes

Here are a few common mistakes that students make when simplifying expressions with the imaginary unit:

  • Forgetting to use the distributive property of multiplication over addition.
  • Not using the properties of the imaginary unit correctly.
  • Not simplifying the expression fully.

Tips and Tricks

Here are a few tips and tricks that can help you simplify expressions with the imaginary unit:

  • Make sure to use the distributive property of multiplication over addition correctly.
  • Use the properties of the imaginary unit correctly.
  • Simplify the expression fully before giving the final answer.

Conclusion

Q: What is the imaginary unit?

A: The imaginary unit, denoted by ii, is a mathematical concept that is used to extend the real number system to the complex number system. It is defined as the square root of −1-1, i.e., i=−1i = \sqrt{-1}.

Q: What are the properties of the imaginary unit?

A: The imaginary unit has a number of important properties, including:

  • i2=−1i^2 = -1
  • i3=−ii^3 = -i
  • i4=1i^4 = 1

Q: How do I simplify expressions with the imaginary unit?

A: To simplify expressions with the imaginary unit, you can use the distributive property of multiplication over addition and the properties of the imaginary unit. Here are the steps to follow:

  1. Rewrite the expression using the distributive property of multiplication over addition.
  2. Simplify the expression by using the properties of the imaginary unit.
  3. Simplify the expression fully before giving the final answer.

Q: What are some common mistakes to avoid when simplifying expressions with the imaginary unit?

A: Here are a few common mistakes to avoid when simplifying expressions with the imaginary unit:

  • Forgetting to use the distributive property of multiplication over addition.
  • Not using the properties of the imaginary unit correctly.
  • Not simplifying the expression fully.

Q: How do I handle expressions with multiple imaginary units?

A: When handling expressions with multiple imaginary units, you can use the distributive property of multiplication over addition and the properties of the imaginary unit. Here are the steps to follow:

  1. Rewrite the expression using the distributive property of multiplication over addition.
  2. Simplify the expression by using the properties of the imaginary unit.
  3. Simplify the expression fully before giving the final answer.

Q: Can I use the imaginary unit to simplify expressions with real numbers?

A: Yes, you can use the imaginary unit to simplify expressions with real numbers. However, you need to be careful when using the imaginary unit with real numbers, as it can lead to complex numbers.

Q: How do I simplify expressions with complex numbers?

A: To simplify expressions with complex numbers, you can use the distributive property of multiplication over addition and the properties of the imaginary unit. Here are the steps to follow:

  1. Rewrite the expression using the distributive property of multiplication over addition.
  2. Simplify the expression by using the properties of the imaginary unit.
  3. Simplify the expression fully before giving the final answer.

Q: What are some examples of expressions that can be simplified using the imaginary unit?

A: Here are a few examples of expressions that can be simplified using the imaginary unit:

  • 5i⋅(−i)5i \cdot (-i)
  • 3i⋅(2i)3i \cdot (2i)
  • 4i⋅(−3i)4i \cdot (-3i)

Q: Can I use the imaginary unit to solve equations with real numbers?

A: Yes, you can use the imaginary unit to solve equations with real numbers. However, you need to be careful when using the imaginary unit with real numbers, as it can lead to complex numbers.

Q: How do I use the imaginary unit to solve equations with complex numbers?

A: To use the imaginary unit to solve equations with complex numbers, you can use the distributive property of multiplication over addition and the properties of the imaginary unit. Here are the steps to follow:

  1. Rewrite the equation using the distributive property of multiplication over addition.
  2. Simplify the equation by using the properties of the imaginary unit.
  3. Solve the equation fully before giving the final answer.

Conclusion

In conclusion, the imaginary unit is a powerful tool that can be used to simplify expressions and solve equations with real and complex numbers. By following the steps outlined in this article and using the tips and tricks provided, you can simplify expressions and solve equations with confidence. Remember to always use the distributive property of multiplication over addition and the properties of the imaginary unit correctly, and to simplify the expression fully before giving the final answer.