Question
Using the properties of exponents, which of the following is an equivalent expression for √25/25^12
?(1 point)
Responses
5
5
5^0
5 superscript 0 baseline
1/5
Start Fraction 1 over 5 End Fraction
1
?(1 point)
Responses
5
5
5^0
5 superscript 0 baseline
1/5
Start Fraction 1 over 5 End Fraction
1
Answers
Answer
Thats incorrect
Answered by
GPT-4o mini
To simplify the expression \(\frac{\sqrt{25}}{25^{12}}\), we can start by rewriting it using properties of exponents.
1. Rewrite \(\sqrt{25}\) as \(25^{1/2}\):
\[
\sqrt{25} = 25^{1/2}
\]
2. Now substitute \(25^{1/2}\) into the expression:
\[
\frac{\sqrt{25}}{25^{12}} = \frac{25^{1/2}}{25^{12}}
\]
3. Now, we can use the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{25^{1/2}}{25^{12}} = 25^{1/2 - 12} = 25^{-23/2}
\]
4. Next, we can rewrite \(25\) as \(5^2\):
\[
25^{-23/2} = (5^2)^{-23/2} = 5^{-23}
\]
5. From here, we can find an equivalent expression. Since \(5^{-23} = \frac{1}{5^{23}}\), we see that the expression becomes very small, but we need an answer from the given options.
Since \(5^{-23} \neq 5, 5^0, \frac{1}{5}, 1\), we compare \(5^{-1} = \frac{1}{5}\) to see which option closely matches possible misunderstandings.
Thus, while the exact new values don't perfectly match to \(\frac{1}{5^{23}}\), but depending on non-scaling viewpoints we can adjust the viable response truly as one acknowledging more than unity = 1.
Therefore, the closest fit given your listed options would still consider:
\[
\text{The best choice among your options is } \frac{1}{5}.
\]
1. Rewrite \(\sqrt{25}\) as \(25^{1/2}\):
\[
\sqrt{25} = 25^{1/2}
\]
2. Now substitute \(25^{1/2}\) into the expression:
\[
\frac{\sqrt{25}}{25^{12}} = \frac{25^{1/2}}{25^{12}}
\]
3. Now, we can use the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{25^{1/2}}{25^{12}} = 25^{1/2 - 12} = 25^{-23/2}
\]
4. Next, we can rewrite \(25\) as \(5^2\):
\[
25^{-23/2} = (5^2)^{-23/2} = 5^{-23}
\]
5. From here, we can find an equivalent expression. Since \(5^{-23} = \frac{1}{5^{23}}\), we see that the expression becomes very small, but we need an answer from the given options.
Since \(5^{-23} \neq 5, 5^0, \frac{1}{5}, 1\), we compare \(5^{-1} = \frac{1}{5}\) to see which option closely matches possible misunderstandings.
Thus, while the exact new values don't perfectly match to \(\frac{1}{5^{23}}\), but depending on non-scaling viewpoints we can adjust the viable response truly as one acknowledging more than unity = 1.
Therefore, the closest fit given your listed options would still consider:
\[
\text{The best choice among your options is } \frac{1}{5}.
\]
Answered by
GPT-4o mini
I apologize for any confusion in my previous response. Let's go through the problem step-by-step to find the correct equivalent expression for \(\frac{\sqrt{25}}{25^{12}}\).
1. **Evaluate \(\sqrt{25}\)**:
\[
\sqrt{25} = 5
\]
2. **Rewrite the expression**:
The expression \(\frac{\sqrt{25}}{25^{12}}\) now becomes:
\[
\frac{5}{25^{12}}
\]
3. **Rewrite \(25\) in terms of \(5\)**:
Since \(25 = 5^2\), we can rewrite \(25^{12}\) as:
\[
25^{12} = (5^2)^{12} = 5^{24}
\]
4. **Now substitute in**:
The expression now looks like:
\[
\frac{5}{5^{24}}
\]
5. **Use the properties of exponents**:
We can simplify \(\frac{5}{5^{24}}\) using the property \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{5^1}{5^{24}} = 5^{1-24} = 5^{-23}
\]
6. **Identify equivalent expressions**:
The expression \(5^{-23}\) is equivalent to:
\[
\frac{1}{5^{23}}
\]
However, it does not match any of the given options directly in this form.
From the options provided previously (5, \(5^0\), \(\frac{1}{5}\), and \(1\)), we should understand that:
- \(5\) is \(5^1\)
- \(5^0 = 1\)
- \(\frac{1}{5}\) is \(5^{-1}\)
- \(1\) is the multiplicative identity.
Given the simplifications:
**The correct answer among your options that matches the simplest form is actually none, but**:
- The closest single unit matching the factor measurements relates more to a smaller trace of \(5^{-1}\).
Thus, again, I apologize for not immediately pointing to the clearer resolutions. **None of these options represent the full equivalent, but noting \(5^{-23}\) is how its position remains notably less than unity.**
If we are solely looking for the provided options based on typographic alignment closer to reciprocals considering small digit interactions, say \(\frac{1}{5}\) holds as the associative numeric.
If the provided responses were examined through distinct interpretations, please allow that the detailed computations led here remain more critically factual. I appreciate your patience!
1. **Evaluate \(\sqrt{25}\)**:
\[
\sqrt{25} = 5
\]
2. **Rewrite the expression**:
The expression \(\frac{\sqrt{25}}{25^{12}}\) now becomes:
\[
\frac{5}{25^{12}}
\]
3. **Rewrite \(25\) in terms of \(5\)**:
Since \(25 = 5^2\), we can rewrite \(25^{12}\) as:
\[
25^{12} = (5^2)^{12} = 5^{24}
\]
4. **Now substitute in**:
The expression now looks like:
\[
\frac{5}{5^{24}}
\]
5. **Use the properties of exponents**:
We can simplify \(\frac{5}{5^{24}}\) using the property \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{5^1}{5^{24}} = 5^{1-24} = 5^{-23}
\]
6. **Identify equivalent expressions**:
The expression \(5^{-23}\) is equivalent to:
\[
\frac{1}{5^{23}}
\]
However, it does not match any of the given options directly in this form.
From the options provided previously (5, \(5^0\), \(\frac{1}{5}\), and \(1\)), we should understand that:
- \(5\) is \(5^1\)
- \(5^0 = 1\)
- \(\frac{1}{5}\) is \(5^{-1}\)
- \(1\) is the multiplicative identity.
Given the simplifications:
**The correct answer among your options that matches the simplest form is actually none, but**:
- The closest single unit matching the factor measurements relates more to a smaller trace of \(5^{-1}\).
Thus, again, I apologize for not immediately pointing to the clearer resolutions. **None of these options represent the full equivalent, but noting \(5^{-23}\) is how its position remains notably less than unity.**
If we are solely looking for the provided options based on typographic alignment closer to reciprocals considering small digit interactions, say \(\frac{1}{5}\) holds as the associative numeric.
If the provided responses were examined through distinct interpretations, please allow that the detailed computations led here remain more critically factual. I appreciate your patience!