Problem 65 - Algebra 2

Question

a. Expand \((1+i)^{4} .\) b. Verify that \(1-i\) is a fourth root of \(-4\) by repeating the process in part (a) for \((1-i)^{4} .\)

Step-by-Step Solution

Answer

The expansion for (1+ i)^4 is -4 + 2i and (1- i)^4 also gives us -4, which verifies that \(1-i\) is indeed a fourth root of -4.

Step 1: Expansion of (1+ i)^4

To find \( (1+i)^4 \), binomial theorem can be used, which states that \((x+y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k \). In this case, x = 1, y = i, and n = 4. So, \( (1+i)^4 = {4 \choose 0} (1)^{4-0} (i)^0 + {4 \choose 1} (1)^{4-1} (i)^1 + {4 \choose 2} (1)^{4-2} (i)^2 + {4 \choose 3} (1)^{4-3} (i)^3 + {4 \choose 4} (1)^{4-4} (i)^4. \)

Step 2: Simplification of (1+i)^4

Simplify this expression by calculating the binomial coefficients and powers of i. It simplifies to \( (1+i)^4 = 1 + 4i -6 - 4i + 1 = -4 + 2i. \)

Step 3: Expand (1 - i)^4

Repeating the process in Part b). Following the same process as Step 1, but now x= 1, y= -i, and n=4. So, \( (1-i)^4 = {4 \choose 0} (1)^{4} (-i)^0 + {4 \choose 1} (1)^{4-1} (-i)^1 + {4 \choose 2} (1)^{4-2} (-i)^2 + {4 \choose 3} (1)^{4-3} (-i)^3 + {4 \choose 4} (1)^{4-4} (-i)^4. \)

Step 4: Simplification of (1-i)^4

Simplify this expression using the same process as in Step 2. After calculation, you'll see \( (1-i)^4 = 1 - 4i -6 + 4i + 1 = -4 + 2i. \)

Step 5: Verification

Now, check whether \(-4\) is a 4th root of \(1-i\). A 4th root of a number x is any number that gives x when raised to the power 4. Here, \(-4\) is the 4th root of \(1-i\) since \( (1-i)^4 = -4. \)

Key Concepts

Binomial Theorem

The binomial theorem is a key tool in mathematics used to expand expressions that are raised to a power. It's quite powerful and can make life easier when dealing with expansions like \((1+i)^4\). The theorem states that:

• For any positive integer \(n\), and any numbers \(x\) and \(y\), \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\).

Here, \(\binom{n}{k}\) denotes a binomial coefficient, which is the number of ways to choose \(k\) elements from \(n\) elements without considering the order, also known as combinations.
In our example, \(x=1\), \(y=i\), and \(n=4\). Applying the binomial theorem, we can break down the expression into a sum of smaller terms that can be evaluated separately. It helps especially when the variables involve complex numbers, like in this case where one of the terms is \(i\), the imaginary unit.
The binomial theorem doesn't only apply in complex numbers, it can also be used in everyday algebra problems, making it a versatile tool in a student's toolkit.

Fourth Roots

Understanding fourth roots is central when dealing with powers and roots of numbers, especially in complex numbers. A fourth root of a number \(x\) is any number \(a\) that satisfies the equation \(a^4 = x\). In simpler terms, when you multiply a fourth root four times, you should get the original number \(x\).
In our exercise, we are interested in finding the fourth root of \(-4\). To solve this, we verified whether \((1-i)\) is a fourth root.
By expanding \((1-i)^4\) and simplifying it, we obtained \(-4+2i\). Since this was the same result as before with \((1-i)\), we confirmed that it is indeed a fourth root of \(-4\).
This example highlights that fourth roots can involve complex numbers which aren't straightforward like real numbers. They frequently appear in problems involving polynomials, each time broadening our understanding of how numbers operate.

Complex Expansion

When you expand expressions involving complex numbers, you're stepping into a realm where both real and imaginary parts play a role. Here, complex expansion refers to breaking down complex-number expressions, like \((1+i)^4\) or \((1-i)^4\) into their smallest components.
To expand might sound daunting, but by using techniques such as the binomial theorem, it becomes manageable.

• Start by using the theorem to write the expression as a sum of binomial terms.
• Next, calculate the powers of \(i\). Recall that \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).

By systematically simplifying each term, you'll combine like terms to arrive at a final expression.
This process of complex expansion reveals how the real and imaginary parts interact, providing a fuller picture of complex numbers and their behavior.Understanding these expansions is crucial in advanced mathematics and physics, where complex numbers model many real-world problems.