Simplify 1/(i^-25)
The question is asking to simplify the mathematical expression given by "1/(i^-25)". It requires you to apply knowledge of complex numbers and properties of exponents, particularly focusing on the imaginary unit "i" which is defined as the square root of -1. Here "i^-25" denotes i raised to the power of -25, and the goal is to simplify this expression to a form without exponents or involving only real numbers, if possible. The solution would involve manipulating the expression according to the laws of exponents and the characteristic cycling of powers of the imaginary unit.
$\frac{1}{i^{- 25}}$
Answer
Solution:
Step 1:
Apply the reciprocal of a negative exponent rule: $\frac{1}{a^{-n}} = a^n$. Thus, we get $i^{25}$.
Step 2:
Express $i^{25}$ as $(i^4)^6 \cdot i$.
Step 2.1:
Separate out $i^{24}$ as $i^{24} \cdot i$.
Step 2.2:
Represent $i^{24}$ as $(i^4)^6$. So, we have $(i^4)^6 \cdot i$.
Step 3:
Recognize that $i^4 = 1$.
Step 3.1:
Decompose $i^4$ as $(i^2)^2$. We now have $((i^2)^2)^6 \cdot i$.
Step 3.2:
Substitute $i^2$ with $-1$. This gives us $((-1)^2)^6 \cdot i$.
Step 3.3:
Compute $(-1)^2$ to get $1$. Our expression simplifies to $1^6 \cdot i$.
Step 4:
Any number raised to the power of 0 is 1. Therefore, $1^6 = 1$.
Step 5:
Finally, multiply $i$ by 1 to obtain the simplified result: $i$.
Knowledge Notes:
The problem involves simplifying an expression with a complex number, specifically the imaginary unit $i$, which is defined as $i = \sqrt{-1}$. Here are the relevant knowledge points:
Negative Exponent Rule: For any non-zero number $a$ and positive integer $n$, $a^{-n} = \frac{1}{a^n}$. This rule is used to move a term with a negative exponent from the denominator to the numerator.
Powers of $i$: The imaginary unit $i$ has the property that $i^2 = -1$. Consequently, powers of $i$ repeat in a cycle: $i^3 = i^2 \cdot i = -i$, $i^4 = (i^2)^2 = 1$, and so on. For any integer $n$, $i^{4n} = 1$ because it is a multiple of $i^4$.
Exponent Laws: When multiplying like bases, we add the exponents: $a^m \cdot a^n = a^{m+n}$. When raising a power to a power, we multiply the exponents: $(a^m)^n = a^{m \cdot n}$.
Simplifying Complex Expressions: When simplifying expressions involving $i$, it's often helpful to reduce the exponent on $i$ to the smallest positive equivalent by using the properties of $i$ mentioned above.
Multiplication by One: Multiplying any number by one does not change the value of that number. This is known as the multiplicative identity property.
By understanding these concepts, the simplification of the expression $1/(i^{-25})$ is achieved by applying the negative exponent rule, recognizing the cyclical nature of powers of $i$, and simplifying the expression step by step.
