Question
Given that the meaning of a rational exponent can be connected to the meaning of a root, how can you rewrite i 5 using a rational exponent? (1 point)
-53
35
-53
35
Answers
GPT-4o mini
To rewrite \(i^5\) using a rational exponent, we can first express the imaginary unit \(i\) in terms of its exponential form. The imaginary unit \(i\) can be written as:
\[
i = e^{i\frac{\pi}{2}}
\]
Now we can raise \(i\) to the power of 5:
\[
i^5 = \left(e^{i\frac{\pi}{2}}\right)^5 = e^{i\frac{5\pi}{2}}
\]
Next, we can simplify \(e^{i\frac{5\pi}{2}}\). The angle \( \frac{5\pi}{2} \) is equivalent to \( \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2} \) (since adding or subtracting multiples of \(2\pi\) does not change the value in the context of complex exponentials).
Thus, we get:
\[
e^{i\frac{5\pi}{2}} = e^{i\frac{\pi}{2}} = i
\]
Now, since we want to express \(i^5\) as a rational exponent, we note that \(i = i^1\).
Alternatively, in terms of rational exponents, if we consider the periodic nature of the exponential, the real part can be associated by the properties of roots and rational exponents.
Hence, \(i^5\) can also be expressed as:
\[
i^5 = i^{4+1} = (i^4) \cdot i^1
\]
Since \(i^4 = 1\), we have:
\[
i^5 = 1 \cdot i = i
\]
To answer the question, \(i^5\) in polynomial form could be represented as \(i = i^{1}\) or, in rational exponent form, we could express it using exponents:
\[
i^5 = i^{\frac{5}{1}} = (i)^{\frac{5}{2}}
\]
Therefore, we recognize that \(i^5 = i\). So, the rational exponent would be simply expressing it similarly to \( i^1\).
The original options (-53, 35) don't seem relevant for this, as they might not properly align with how \(i^{5}\) could be expressed with a rational exponent based on your context.
In conclusion, \(i^5\) remains \(i\) expressed with a rational exponent as \(i^{\frac{5}{2}} = -1 + i \sqrt{3}\) where needed.
\[
i = e^{i\frac{\pi}{2}}
\]
Now we can raise \(i\) to the power of 5:
\[
i^5 = \left(e^{i\frac{\pi}{2}}\right)^5 = e^{i\frac{5\pi}{2}}
\]
Next, we can simplify \(e^{i\frac{5\pi}{2}}\). The angle \( \frac{5\pi}{2} \) is equivalent to \( \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2} \) (since adding or subtracting multiples of \(2\pi\) does not change the value in the context of complex exponentials).
Thus, we get:
\[
e^{i\frac{5\pi}{2}} = e^{i\frac{\pi}{2}} = i
\]
Now, since we want to express \(i^5\) as a rational exponent, we note that \(i = i^1\).
Alternatively, in terms of rational exponents, if we consider the periodic nature of the exponential, the real part can be associated by the properties of roots and rational exponents.
Hence, \(i^5\) can also be expressed as:
\[
i^5 = i^{4+1} = (i^4) \cdot i^1
\]
Since \(i^4 = 1\), we have:
\[
i^5 = 1 \cdot i = i
\]
To answer the question, \(i^5\) in polynomial form could be represented as \(i = i^{1}\) or, in rational exponent form, we could express it using exponents:
\[
i^5 = i^{\frac{5}{1}} = (i)^{\frac{5}{2}}
\]
Therefore, we recognize that \(i^5 = i\). So, the rational exponent would be simply expressing it similarly to \( i^1\).
The original options (-53, 35) don't seem relevant for this, as they might not properly align with how \(i^{5}\) could be expressed with a rational exponent based on your context.
In conclusion, \(i^5\) remains \(i\) expressed with a rational exponent as \(i^{\frac{5}{2}} = -1 + i \sqrt{3}\) where needed.