Question
What rational number could be used as an exponent to rewrite 4 divided by 23
Answers
Answered by
GPT-4o mini
To rewrite \( \frac{4}{23} \) using a rational number as an exponent, we can express both the numerator and denominator as powers of a common base.
One common base to use is 2, since \( 4 \) can be written as \( 2^2 \). We can rewrite \( \frac{4}{23} \) as:
\[
\frac{4}{23} = \frac{2^2}{23}
\]
Now, to express it in terms of a rational exponent, we can use the property of exponents that states:
\[
\frac{a^m}{b^n} = a^m \cdot b^{-n}
\]
In our case, we can find a way to express this as a power:
Let’s denote \( 23 \) as \( 23^1 \), then we can rewrite \( \frac{2^2}{23} \) as:
\[
2^2 \cdot 23^{-1}
\]
Thus, we have:
\[
\frac{4}{23} = 2^2 \cdot 23^{-1}
\]
If we want to explicitly write \( \frac{4}{23} \) as a single power, we can take the logarithm of both \( 4 \) and \( 23 \) on some common base, but typically the expression is best seen as it remains as separate bases.
In general, if you wish to express this functionally and at once, the relationship could be made clearer that \( \frac{4}{23} = 2^{2} \cdot 23^{-1} \), where the exponent of \( 23 \) is \(-1\).
One common base to use is 2, since \( 4 \) can be written as \( 2^2 \). We can rewrite \( \frac{4}{23} \) as:
\[
\frac{4}{23} = \frac{2^2}{23}
\]
Now, to express it in terms of a rational exponent, we can use the property of exponents that states:
\[
\frac{a^m}{b^n} = a^m \cdot b^{-n}
\]
In our case, we can find a way to express this as a power:
Let’s denote \( 23 \) as \( 23^1 \), then we can rewrite \( \frac{2^2}{23} \) as:
\[
2^2 \cdot 23^{-1}
\]
Thus, we have:
\[
\frac{4}{23} = 2^2 \cdot 23^{-1}
\]
If we want to explicitly write \( \frac{4}{23} \) as a single power, we can take the logarithm of both \( 4 \) and \( 23 \) on some common base, but typically the expression is best seen as it remains as separate bases.
In general, if you wish to express this functionally and at once, the relationship could be made clearer that \( \frac{4}{23} = 2^{2} \cdot 23^{-1} \), where the exponent of \( 23 \) is \(-1\).