Key Concepts and Formulas

• Complex Number Division: To simplify a complex fraction $\frac{a+bi}{c+di}$, multiply the numerator and denominator by the conjugate of the denominator, $c-di$. This utilizes the property $(c+di)(c-di) = c^2 + d^2$.
• Powers of the Imaginary Unit ($i$): Recall that $i = \sqrt{-1}$, so $i^2 = -1$. The powers of $i$ cycle: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and so on. $i^{4n} = 1$, $i^{4n+1} = i$, $i^{4n+2} = -1$, $i^{4n+3} = -i$ for any integer $n$.

Step-by-Step Solution

Let's analyze the given equation: ${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$

Step 1: Simplify the Base Complex Number

Our first objective is to simplify the complex fraction $\frac{1+i}{1-i}$ inside the parenthesis.

Why this step? Simplifying the base will transform the complex fraction into a simpler form, making it easier to raise to a power.

To simplify, we multiply the numerator and the denominator by the conjugate of the denominator. The denominator is $1-i$, so its conjugate is $1+i$.

${\left[ {\frac{{\left( {1 + i} \right)}}{{\left( {1 - i} \right)}} \times \frac{{\left( {1 + i} \right)}}{{\left( {1 + i} \right)}}} \right]^x} = 1$

Now, let's perform the multiplication for the numerator and the denominator separately:

• Numerator: We have $(1+i)(1+i) = (1+i)^2$. Using the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$: $(1+i)^2 = 1^2 + 2(1)(i) + i^2$ Since $i^2 = -1$, substitute this value: $1 + 2i - 1 = 2i$

• Denominator: We have $(1-i)(1+i)$. Using the algebraic identity $(a-b)(a+b) = a^2 - b^2$: $(1-i)(1+i) = 1^2 - i^2$ Since $i^2 = -1$, substitute this value: $1 - (-1) = 1 + 1 = 2$

Now, substitute these simplified expressions back into the equation: ${\left[ {\frac{{2i}}{2}} \right]^x} = 1$

Why this step? By applying standard algebraic identities and the definition $i^2=-1$, we've eliminated the imaginary part from the denominator and simplified the numerator.

Further simplify the fraction: ${(i)^x} = 1$

We have now reduced the original complex equation to a much simpler form involving only powers of $i$.

Step 2: Determine Possible Values for $x$ using Powers of $i$

We need to find the values of $x$ for which $i^x = 1$.

Why this step? Understanding the periodic nature of powers of $i$ is fundamental to solving this problem.

Recall the cyclic pattern of powers of $i$: $i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$. In general, $i^{4n}=1$ for any integer $n$.

Therefore, $x$ must be of the form $4n$, where $n$ is an integer.

Step 3: Compare with Given Options

We have determined that $x$ must be of the form $4n$, where $n$ is a positive integer. Let's examine each option, keeping in mind that $n$ is a positive integer (i.e., $n \in \{1, 2, 3, \dots\}$):

• (A) $x = 2n + 1$: This form represents odd numbers. For example, if $n=1$, then $x=3$ and $i^3 = -i \neq 1$. This option is incorrect.

• (B) $x = 4n$: This form represents positive multiples of 4. For example, if $n=1$, then $x=4$ and $i^4 = 1$. This option correctly describes all possible values of $x$.

• (C) $x = 2n$: This form represents positive even numbers. For example, if $n=1$, then $x=2$ and $i^2 = -1 \neq 1$. This option is incorrect.

• (D) $x = 4n + 1$: This form represents numbers that leave a remainder of 1 when divided by 4. For example, if $n=1$, then $x=5$ and $i^5 = i \neq 1$. This option is incorrect.

Therefore, $x = 4n$ where n is a positive integer. However, the provided correct answer is (A). Let us assume there is a typo in the options, and the correct answer is $x=4n$. In this case, the correct option is (B).

Common Mistakes & Tips

• Simplify the base first: Always simplify the complex fraction before raising it to a power.
• Remember the powers of $i$: Know the cyclic pattern of $i^1, i^2, i^3, i^4$.
• Double-check calculations: Be careful with signs, especially when dealing with $i^2 = -1$.

Summary

To solve the equation ${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$, we first simplify the base to get $i^x = 1$. Then, we use the properties of powers of $i$ to find that $x$ must be a multiple of 4. This leads to the solution $x = 4n$, where $n$ is a positive integer, which corresponds to option (B).

The final answer is \boxed{4n}, which corresponds to option (B).