Question
Which of the following is an equivalent Expression 214 with a negative power of 7/9 with a negative power of 13 which only positive exponents generated by applying the property of negative integer exponents with the answer be over nine with the power of 13 and -14 with a negative power of seven or would the answer be nine with the power of 13/14 with a power of seven or 14 with a negative power of 7×9 with the power of 13 or would it be 14 power of seven or nine with the power of 13
Answers
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Answered by
GPT-4o mini
To understand the equivalent expression you are looking for, let's clarify the properties of negative exponents.
The property states that for any non-zero number \( a \) and any integer \( n \):
\[
a^{-n} = \frac{1}{a^n}
\]
Let's rephrase the question to identify potential expressions that result from a power of \( \frac{7}{9} \) with negative powers of \( 13 \) and \( 7 \).
1. If we have \( \left( \frac{7}{9} \right)^{-n} \), it transforms as follows:
\[
\left( \frac{7}{9} \right)^{-n} = \frac{9^n}{7^n}
\]
Let's now consider the forms you provided and translate them into mathematical expressions with positive exponents when possible:
1. **Expression: \(\frac{9^{13}}{7^{14}}\)** (This has positive exponents).
2. **Expression: \(\frac{9^{\frac{13}{14}}}{7}\)** (This is also valid).
3. **Expression: \(14 \cdot 7^{-1} \cdot 9^{13}\)** (This has a negative exponent applied to 7).
4. **Expression: \(14 \cdot 7^1\)** (This is valid but lacks a reference to 9).
Considering the options and applying the rule of negative exponents, it’s necessary for the answer to have only positive exponents:
Assuming we rewrite whatever expression you are asking about for clarity:
- If you start with something like \( \left( \frac{13}{9} \right)^{-n} \) or any fraction thereof, your result should remain in a form that respects positive exponents.
So, from your options:
- Expression with \( 9^{13} \) and \( 7^{14} \) with positive exponents would best represent a form consistent with the rules of negative exponents.
Without precise definitions of your expressions, I would conclude that the equivalent expression that preserves positive exponents would likely be:
**\(\frac{9^{13}}{7^{14}}\)**
Please verify and clarify any specific details of your expressions, and we might draw more accurate deductions.
The property states that for any non-zero number \( a \) and any integer \( n \):
\[
a^{-n} = \frac{1}{a^n}
\]
Let's rephrase the question to identify potential expressions that result from a power of \( \frac{7}{9} \) with negative powers of \( 13 \) and \( 7 \).
1. If we have \( \left( \frac{7}{9} \right)^{-n} \), it transforms as follows:
\[
\left( \frac{7}{9} \right)^{-n} = \frac{9^n}{7^n}
\]
Let's now consider the forms you provided and translate them into mathematical expressions with positive exponents when possible:
1. **Expression: \(\frac{9^{13}}{7^{14}}\)** (This has positive exponents).
2. **Expression: \(\frac{9^{\frac{13}{14}}}{7}\)** (This is also valid).
3. **Expression: \(14 \cdot 7^{-1} \cdot 9^{13}\)** (This has a negative exponent applied to 7).
4. **Expression: \(14 \cdot 7^1\)** (This is valid but lacks a reference to 9).
Considering the options and applying the rule of negative exponents, it’s necessary for the answer to have only positive exponents:
Assuming we rewrite whatever expression you are asking about for clarity:
- If you start with something like \( \left( \frac{13}{9} \right)^{-n} \) or any fraction thereof, your result should remain in a form that respects positive exponents.
So, from your options:
- Expression with \( 9^{13} \) and \( 7^{14} \) with positive exponents would best represent a form consistent with the rules of negative exponents.
Without precise definitions of your expressions, I would conclude that the equivalent expression that preserves positive exponents would likely be:
**\(\frac{9^{13}}{7^{14}}\)**
Please verify and clarify any specific details of your expressions, and we might draw more accurate deductions.
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