Question
Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3 rasied to the power of −17/8 rasied to the power of −14 with positive exponents only.(1 point) 3 rasied to the power of −17/8 rasied to the power of −14=
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Answer
i dont under stand the answer could you do it in word form
Answered by
GPT-4o mini
To apply the property of negative integer exponents and rewrite the expression \(3^{-\frac{17}{8}}^{-14}\) with positive exponents, we can use the rule that states \(a^{-b} = \frac{1}{a^b}\).
Here's how we can transform the given expression step by step:
1. **Rewrite the expression using the negative exponent property**:
\[
3^{-\frac{17}{8}}^{-14} = 3^{\left(-\frac{17}{8} \cdot -14\right)}
\]
2. **Multiply the exponents**:
\[
-\frac{17}{8} \cdot -14 = \frac{17 \cdot 14}{8} = \frac{238}{8}
\]
3. **Simplify the fraction**:
\[
\frac{238}{8} = \frac{119}{4}
\]
4. **Rewrite with positive exponents**:
Therefore, we have:
\[
3^{-\frac{17}{8}}^{-14} = 3^{\frac{119}{4}}
\]
So, the expression equivalent to \(3^{-\frac{17}{8}}^{-14}\) with positive exponents only is:
\[
3^{\frac{119}{4}}
\]
Here's how we can transform the given expression step by step:
1. **Rewrite the expression using the negative exponent property**:
\[
3^{-\frac{17}{8}}^{-14} = 3^{\left(-\frac{17}{8} \cdot -14\right)}
\]
2. **Multiply the exponents**:
\[
-\frac{17}{8} \cdot -14 = \frac{17 \cdot 14}{8} = \frac{238}{8}
\]
3. **Simplify the fraction**:
\[
\frac{238}{8} = \frac{119}{4}
\]
4. **Rewrite with positive exponents**:
Therefore, we have:
\[
3^{-\frac{17}{8}}^{-14} = 3^{\frac{119}{4}}
\]
So, the expression equivalent to \(3^{-\frac{17}{8}}^{-14}\) with positive exponents only is:
\[
3^{\frac{119}{4}}
\]
Answered by
GPT-4o mini
Of course! Let’s break it down step by step in a more detailed way using words.
1. **Start with the expression**: We have the expression where we are looking at the number 3 raised to the power of negative \( \frac{17}{8} \) raised to the power of negative 14.
2. **Understand the rules about negative exponents**: A negative exponent means we take the reciprocal. For example, \( a^{-b} \) is equal to \( \frac{1}{a^b} \).
3. **Apply the negative exponent rule**: When we raise a number to a negative exponent, we can rewrite it as the reciprocal of that number raised to the corresponding positive exponent. So, if we see \( 3^{-\frac{17}{8}} \), it can be rewritten as \( \frac{1}{3^{\frac{17}{8}}} \).
4. **Combine the exponents**: We also have another exponent, which is negative 14. The expression becomes more complicated because we are raising to another exponent. Specifically, when we have something like \( A^{-B} \), it becomes \( \left( A^{-B} \right) = 3^{-\frac{17}{8}} \) raised to the power of negative 14. This means we can multiply the two negative exponents together.
5. **Multiply the two negative exponents**: When we multiply the negative fraction \( -\frac{17}{8} \) by negative 14, we find that the negatives cancel out, resulting in a positive exponent. We multiply \( -\frac{17}{8} \) by -14, which can be thought of as:
- First, multiplying just the numbers: \( 17 \times 14 \) equals 238.
- Then, we divide that by 8, resulting in \( \frac{238}{8} \).
- After simplifying, \( \frac{238}{8} \) gives us \( \frac{119}{4} \).
6. **Rewrite the original expression**: Finally, instead of having two negative exponents, we now have a positive exponent. So, the original expression \( 3^{-\frac{17}{8}}^{-14} \) can now be rewritten simply as \( 3^{\frac{119}{4}} \).
In summary, the original expression \( 3^{-\frac{17}{8}}^{-14} \) can be rewritten in positive exponents as \( 3^{\frac{119}{4}} \).
I hope this explanation makes it clearer! Let me know if you have any other questions.
1. **Start with the expression**: We have the expression where we are looking at the number 3 raised to the power of negative \( \frac{17}{8} \) raised to the power of negative 14.
2. **Understand the rules about negative exponents**: A negative exponent means we take the reciprocal. For example, \( a^{-b} \) is equal to \( \frac{1}{a^b} \).
3. **Apply the negative exponent rule**: When we raise a number to a negative exponent, we can rewrite it as the reciprocal of that number raised to the corresponding positive exponent. So, if we see \( 3^{-\frac{17}{8}} \), it can be rewritten as \( \frac{1}{3^{\frac{17}{8}}} \).
4. **Combine the exponents**: We also have another exponent, which is negative 14. The expression becomes more complicated because we are raising to another exponent. Specifically, when we have something like \( A^{-B} \), it becomes \( \left( A^{-B} \right) = 3^{-\frac{17}{8}} \) raised to the power of negative 14. This means we can multiply the two negative exponents together.
5. **Multiply the two negative exponents**: When we multiply the negative fraction \( -\frac{17}{8} \) by negative 14, we find that the negatives cancel out, resulting in a positive exponent. We multiply \( -\frac{17}{8} \) by -14, which can be thought of as:
- First, multiplying just the numbers: \( 17 \times 14 \) equals 238.
- Then, we divide that by 8, resulting in \( \frac{238}{8} \).
- After simplifying, \( \frac{238}{8} \) gives us \( \frac{119}{4} \).
6. **Rewrite the original expression**: Finally, instead of having two negative exponents, we now have a positive exponent. So, the original expression \( 3^{-\frac{17}{8}}^{-14} \) can now be rewritten simply as \( 3^{\frac{119}{4}} \).
In summary, the original expression \( 3^{-\frac{17}{8}}^{-14} \) can be rewritten in positive exponents as \( 3^{\frac{119}{4}} \).
I hope this explanation makes it clearer! Let me know if you have any other questions.
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