Evaluate:${ I^{11} \cdot I^{13} \cdot I^7 \cdot I^9 }$

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Introduction

In mathematics, the imaginary unit $i$ is defined as the square root of $-1$. It is a fundamental concept in algebra and is used to extend the real number system to the complex number system. When dealing with powers of $i$, it is essential to understand the pattern of its powers to simplify expressions involving $i$. In this article, we will evaluate the expression $i^{11} \cdot i^{13} \cdot i^7 \cdot i^9$ and explore the properties of the imaginary unit.

Properties of the Imaginary Unit

The imaginary unit $i$ has a repeating pattern of powers:

$i^1 = i$ $i^2 = -1$ $i^3 = -i$ $i^4 = 1$ $i^5 = i$ $i^6 = -1$ $i^7 = -i$ $i^8 = 1$ $i^9 = i$ $i^{10} = -1$ $i^{11} = -i$ $i^{12} = -1$ $i^{13} = i$ $i^{14} = 1$

As we can see, the powers of $i$ repeat every four powers. This means that we can simplify any power of $i$ by finding the remainder when the exponent is divided by 4.

Simplifying the Expression

To simplify the expression $i^{11} \cdot i^{13} \cdot i^7 \cdot i^9$, we can use the properties of the imaginary unit. We can rewrite each power of $i$ in terms of its remainder when divided by 4:

$i^{11} = i^{(4 \cdot 2 + 3)} = i^3 = -i$ $i^{13} = i^{(4 \cdot 3 + 1)} = i^1 = i$ $i^7 = i^{(4 \cdot 1 + 3)} = i^3 = -i$ $i^9 = i^{(4 \cdot 2 + 1)} = i^1 = i$

Now, we can multiply the simplified powers of $i$:

$i^{11} \cdot i^{13} \cdot i^7 \cdot i^9 = (-i) \cdot i \cdot (-i) \cdot i$

Multiplying Powers of $i$

When multiplying powers of $i$, we can use the fact that $i^2 = -1$. This means that any even power of $i$ can be simplified to $-1$. For example:

$i^2 = (-1)$ $i^4 = (i^2)^2 = (-1)^2 = 1$ $i^6 = (i^2)^3 = (-1)^3 = -1$ $i^8 = (i^2)^4 = (-1)^4 = 1$

Using this property, we can simplify the expression:

$(-i) \cdot i \cdot (-i) \cdot i = (-i^2) \cdot (-i^2) = (-(-1)) \cdot (-(-1)) = 1 \cdot 1 = 1$

Conclusion

In this article, we evaluated the expression $i^{11} \cdot i^{13} \cdot i^7 \cdot i^9$ and explored the properties of the imaginary unit. We simplified the expression by rewriting each power of $i$ in terms of its remainder when divided by 4 and then multiplied the simplified powers of $i$. The final result is $1$, which is the product of the four powers of $i$.

Final Answer

The final answer is $\boxed{1}$.

Related Topics

• Properties of the imaginary unit
• Simplifying powers of $i$
• Multiplying powers of $i$
• Complex numbers

References

• [1] "Imaginary Unit" by MathWorld
• [2] "Complex Numbers" by Wolfram MathWorld
• [3] "Properties of the Imaginary Unit" by Purplemath

Note: The references provided are for informational purposes only and are not a substitute for the original sources.

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Introduction

In our previous article, we evaluated the expression $i^{11} \cdot i^{13} \cdot i^7 \cdot i^9$ and explored the properties of the imaginary unit. In this article, we will answer some frequently asked questions related to evaluating powers of $i$.

Q1: What is the pattern of powers of $i$?

A1: The powers of $i$ repeat every four powers:

$i^1 = i$ $i^2 = -1$ $i^3 = -i$ $i^4 = 1$ $i^5 = i$ $i^6 = -1$ $i^7 = -i$ $i^8 = 1$ $i^9 = i$ $i^{10} = -1$ $i^{11} = -i$ $i^{12} = -1$ $i^{13} = i$ $i^{14} = 1$

Q2: How do I simplify a power of $i$?

A2: To simplify a power of $i$, you can find the remainder when the exponent is divided by 4. For example:

$i^{11} = i^{(4 \cdot 2 + 3)} = i^3 = -i$ $i^{13} = i^{(4 \cdot 3 + 1)} = i^1 = i$

Q3: What is the property of $i^2$?

A3: The property of $i^2$ is that it is equal to $-1$. This means that any even power of $i$ can be simplified to $-1$.

Q4: How do I multiply powers of $i$?

A4: When multiplying powers of $i$, you can use the fact that $i^2 = -1$. This means that any even power of $i$ can be simplified to $-1$. For example:

$(-i) \cdot i \cdot (-i) \cdot i = (-i^2) \cdot (-i^2) = (-(-1)) \cdot (-(-1)) = 1 \cdot 1 = 1$

Q5: What is the final answer to the expression $i^{11} \cdot i^{13} \cdot i^7 \cdot i^9$?

A5: The final answer to the expression $i^{11} \cdot i^{13} \cdot i^7 \cdot i^9$ is $\boxed{1}$.

Q6: Can I use the properties of $i$ to simplify other expressions?

A6: Yes, you can use the properties of $i$ to simplify other expressions involving powers of $i$. For example:

$i^{23} = i^{(4 \cdot 5 + 3)} = i^3 = -i$ $i^{29} = i^{(4 \cdot 7 + 1)} = i^1 = i$

Q7: What are some common mistakes to avoid when evaluating powers of $i$?

A7: Some common mistakes to avoid when evaluating powers of $i$ include:

• Not finding the remainder when the exponent is divided by 4
• Not using the property of $i^2 = -1$
• Not simplifying even powers of $i$ to $-1$

Q8: Can I use a calculator to evaluate powers of $i$?

A8: Yes, you can use a calculator to evaluate powers of $i$. However, it is essential to understand the properties of $i$ and how to simplify powers of $i$ manually.

Q9: What are some real-world applications of powers of $i$?

A9: Powers of $i$ have many real-world applications, including:

• Electrical engineering: Powers of $i$ are used to represent alternating current (AC) and alternating voltage (AC).
• Signal processing: Powers of $i$ are used to represent complex signals and filter out noise.
• Computer graphics: Powers of $i$ are used to represent complex numbers and perform geometric transformations.

Q10: Can I use powers of $i$ to solve other mathematical problems?

A10: Yes, you can use powers of $i$ to solve other mathematical problems, including:

• Solving quadratic equations
• Finding the roots of polynomials
• Evaluating trigonometric functions

Conclusion

In this article, we answered some frequently asked questions related to evaluating powers of $i$. We covered topics such as the pattern of powers of $i$, simplifying powers of $i$, multiplying powers of $i$, and real-world applications of powers of $i$. We hope that this article has been helpful in understanding the properties of $i$ and how to evaluate powers of $i$.