Simplify The Expression:b. $(-1+i)^7$

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Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. The expression $(-1+i)^7$ is a good example of a complex expression that requires simplification. In this article, we will explore the steps to simplify this expression and provide a clear understanding of the process.

Understanding Complex Numbers

Before we dive into simplifying the expression, it's essential to understand 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, which satisfies the equation $i^2 = -1$. Complex numbers can be represented on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

Simplifying the Expression

To simplify the expression $(-1+i)^7$, we can use the binomial theorem, which states that for any positive integer $n$, we can expand the expression $(a+b)^n$ as follows:

$$(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n$$

where $\binom{n}{k}$ is the binomial coefficient, which is defined as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Applying the Binomial Theorem

Now, let's apply the binomial theorem to simplify the expression $(-1+i)^7$. We can expand the expression as follows:

$$(-1+i)^7 = \binom{7}{0}(-1)^7 + \binom{7}{1}(-1)^6i + \binom{7}{2}(-1)^5i^2 + \binom{7}{3}(-1)^4i^3 + \binom{7}{4}(-1)^3i^4 + \binom{7}{5}(-1)^2i^5 + \binom{7}{6}(-1)^1i^6 + \binom{7}{7}i^7$$

Evaluating the Binomial Coefficients

Now, let's evaluate the binomial coefficients:

$$\binom{7}{0} = 1$$

$$\binom{7}{1} = 7$$

$$\binom{7}{2} = 21$$

$$\binom{7}{3} = 35$$

$$\binom{7}{4} = 35$$

$$\binom{7}{5} = 21$$

$$\binom{7}{6} = 7$$

$$\binom{7}{7} = 1$$

Simplifying the Expression

Now, let's simplify the expression by substituting the values of the binomial coefficients:

$$(-1+i)^7 = 1 + 7i + 21i^2 + 35i^3 + 35i^4 + 21i^5 + 7i^6 + i^7$$

Evaluating the Powers of $i$

Now, let's evaluate the powers of $i$:

$$i^2 = -1$$

$$i^3 = -i$$

$$i^4 = 1$$

$$i^5 = i$$

$$i^6 = -1$$

$$i^7 = -i$$

Simplifying the Expression

Now, let's simplify the expression by substituting the values of the powers of $i$:

$$(-1+i)^7 = 1 + 7i + 21(-1) + 35(-i) + 35(1) + 21(i) + 7(-1) + (-i)$$

Combining Like Terms

Now, let's combine like terms:

$$(-1+i)^7 = 1 + 7i - 21 - 35i + 35 + 21i - 7 - i$$

Simplifying the Expression

Now, let's simplify the expression by combining like terms:

$$(-1+i)^7 = (1 - 21 + 35 - 7) + (7i - 35i + 21i - i)$$

Evaluating the Expression

Now, let's evaluate the expression:

$$(-1+i)^7 = 8 - 8i$$

Conclusion

In this article, we simplified the expression $(-1+i)^7$ using the binomial theorem. We expanded the expression, evaluated the binomial coefficients, simplified the expression by substituting the values of the powers of $i$, combined like terms, and finally evaluated the expression. The simplified expression is $8 - 8i$. This example demonstrates the importance of simplifying complex expressions in mathematics.

Frequently Asked Questions

• Q: What is the binomial theorem? A: The binomial theorem is a mathematical formula that allows us to expand the expression $(a+b)^n$.
• Q: How do I apply the binomial theorem to simplify an expression? A: To apply the binomial theorem, you need to expand the expression, evaluate the binomial coefficients, simplify the expression by substituting the values of the powers of $i$, combine like terms, and finally evaluate the expression.
• Q: What is the importance of simplifying complex expressions in mathematics? A: Simplifying complex expressions is crucial in mathematics because it helps us solve problems efficiently and understand the underlying concepts.

Further Reading

• Binomial Theorem: A Comprehensive Guide
• Simplifying Complex Expressions: A Step-by-Step Guide
• Mathematics: A Subject that Demands Practice and Patience

References

• [1] Binomial Theorem. (n.d.). In Encyclopedia Britannica.
• [2] Simplifying Complex Expressions. (n.d.). In Math Open Reference.
• [3] Mathematics. (n.d.). In Wikipedia.
  

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Introduction

In our previous article, we simplified the expression $(-1+i)^7$ using the binomial theorem. In this article, we will answer some frequently asked questions related to simplifying complex expressions and provide additional information to help you understand the concept better.

Q&A

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that allows us to expand the expression $(a+b)^n$. It is a powerful tool for simplifying complex expressions and is widely used in mathematics, physics, and engineering.

Q: How do I apply the binomial theorem to simplify an expression?

A: To apply the binomial theorem, you need to follow these steps:

1. Expand the expression using the binomial theorem formula.
2. Evaluate the binomial coefficients.
3. Simplify the expression by substituting the values of the powers of $i$.
4. Combine like terms.
5. Finally, evaluate the expression.

Q: What is the importance of simplifying complex expressions in mathematics?

A: Simplifying complex expressions is crucial in mathematics because it helps us solve problems efficiently and understand the underlying concepts. It also helps us to identify patterns and relationships between different mathematical objects.

Q: How do I simplify complex expressions with negative exponents?

A: To simplify complex expressions with negative exponents, you need to use the rule that $a^{-n} = \frac{1}{a^n}$. This rule allows you to rewrite negative exponents as positive exponents and simplify the expression.

Q: Can I use the binomial theorem to simplify expressions with complex numbers?

A: Yes, you can use the binomial theorem to simplify expressions with complex numbers. The binomial theorem formula works for any complex numbers, and you can use it to simplify expressions with complex numbers.

Q: How do I simplify expressions with multiple complex numbers?

A: To simplify expressions with multiple complex numbers, you need to use the binomial theorem formula multiple times. You can also use the distributive property to simplify the expression.

Q: What are some common mistakes to avoid when simplifying complex expressions?

A: Some common mistakes to avoid when simplifying complex expressions include:

• Not using the binomial theorem formula correctly.
• Not evaluating the binomial coefficients correctly.
• Not simplifying the expression by substituting the values of the powers of $i$.
• Not combining like terms correctly.
• Not evaluating the expression correctly.

Q: Can I use technology to simplify complex expressions?

A: Yes, you can use technology to simplify complex expressions. Many calculators and computer algebra systems (CAS) have built-in functions that can simplify complex expressions.

Q: How do I check my work when simplifying complex expressions?

A: To check your work when simplifying complex expressions, you need to:

• Verify that you have used the binomial theorem formula correctly.
• Verify that you have evaluated the binomial coefficients correctly.
• Verify that you have simplified the expression by substituting the values of the powers of $i$.
• Verify that you have combined like terms correctly.
• Verify that you have evaluated the expression correctly.

Conclusion

In this article, we answered some frequently asked questions related to simplifying complex expressions and provided additional information to help you understand the concept better. We hope that this article has been helpful in clarifying any doubts you may have had about simplifying complex expressions.

Frequently Asked Questions

• Q: What is the binomial theorem? A: The binomial theorem is a mathematical formula that allows us to expand the expression $(a+b)^n$.
• Q: How do I apply the binomial theorem to simplify an expression? A: To apply the binomial theorem, you need to follow these steps: expand the expression, evaluate the binomial coefficients, simplify the expression by substituting the values of the powers of $i$, combine like terms, and finally evaluate the expression.
• Q: What is the importance of simplifying complex expressions in mathematics? A: Simplifying complex expressions is crucial in mathematics because it helps us solve problems efficiently and understand the underlying concepts.

Further Reading

• Binomial Theorem: A Comprehensive Guide
• Simplifying Complex Expressions: A Step-by-Step Guide
• Mathematics: A Subject that Demands Practice and Patience

References

• [1] Binomial Theorem. (n.d.). In Encyclopedia Britannica.
• [2] Simplifying Complex Expressions. (n.d.). In Math Open Reference.
• [3] Mathematics. (n.d.). In Wikipedia.