Question
hey bot
Apply the properties of integer exponents to generate an equivalent algebraic expression to (4x4y−4)3(x7y3)2 .(1 point) Responses x2y1864 Start Fraction x squared y superscript 18 baseline over 64 End Fraction 64x20 Start Fraction 64 over x superscript 20 baseline End Fraction 64x2y18 Start Fraction 64 over x squared y superscript 18 baseline End Fraction 4x2y18 Start Fraction 4 over x squared y superscript 18 baseline End Fraction
Apply the properties of integer exponents to generate an equivalent algebraic expression to (4x4y−4)3(x7y3)2 .(1 point) Responses x2y1864 Start Fraction x squared y superscript 18 baseline over 64 End Fraction 64x20 Start Fraction 64 over x superscript 20 baseline End Fraction 64x2y18 Start Fraction 64 over x squared y superscript 18 baseline End Fraction 4x2y18 Start Fraction 4 over x squared y superscript 18 baseline End Fraction
Answers
Answered by
GPT-4o mini
To simplify the expression \((4x^4y^{-4})^3(x^7y^3)^2\) using the properties of integer exponents, we can follow these steps:
1. **Apply the power to each term in the parentheses:**
\[
(4x^4y^{-4})^3 = 4^3 \cdot (x^4)^3 \cdot (y^{-4})^3 = 64x^{12}y^{-12}
\]
\[
(x^7y^3)^2 = (x^7)^2 \cdot (y^3)^2 = x^{14}y^6
\]
2. **Combine the results:**
Now we have:
\[
64x^{12}y^{-12} \cdot x^{14}y^6
\]
3. **Use the properties of exponents to combine like bases:**
\[
= 64 \cdot x^{12 + 14} \cdot y^{-12 + 6} = 64x^{26}y^{-6}
\]
4. **Rewrite the expression with positive exponents:**
The term \(y^{-6}\) can be rewritten as \(\frac{1}{y^6}\):
\[
= \frac{64x^{26}}{y^6}
\]
Now we look at the answer choices provided based on our simplification.
The equivalent expression is:
\[
\frac{64x^{26}}{y^6}
\]
However, none of the provided answer choices exactly match this result. Please double-check the choices, or if there is a specific requirement for formatting, let me know!
1. **Apply the power to each term in the parentheses:**
\[
(4x^4y^{-4})^3 = 4^3 \cdot (x^4)^3 \cdot (y^{-4})^3 = 64x^{12}y^{-12}
\]
\[
(x^7y^3)^2 = (x^7)^2 \cdot (y^3)^2 = x^{14}y^6
\]
2. **Combine the results:**
Now we have:
\[
64x^{12}y^{-12} \cdot x^{14}y^6
\]
3. **Use the properties of exponents to combine like bases:**
\[
= 64 \cdot x^{12 + 14} \cdot y^{-12 + 6} = 64x^{26}y^{-6}
\]
4. **Rewrite the expression with positive exponents:**
The term \(y^{-6}\) can be rewritten as \(\frac{1}{y^6}\):
\[
= \frac{64x^{26}}{y^6}
\]
Now we look at the answer choices provided based on our simplification.
The equivalent expression is:
\[
\frac{64x^{26}}{y^6}
\]
However, none of the provided answer choices exactly match this result. Please double-check the choices, or if there is a specific requirement for formatting, let me know!
Answer
answer?
Answered by
GPT-4o mini
The simplified expression for \((4x^4y^{-4})^3(x^7y^3)^2\) is:
\[
\frac{64x^{26}}{y^6}
\]
However, from your original options, the closest equivalent expression seems to be:
**64 over x squared y to the 18.** \( \frac{64}{x^2y^{18}} \)
It looks like there might be a misunderstanding or error in the available responses, as the exponential terms and constants do not directly match the fully simplified results we derived.
If you want a direct selection based on your initial options, please recheck the provided options or choose another one based on their relevance, but mathematically, the correct simplification leads to \( \frac{64x^{26}}{y^6} \).
\[
\frac{64x^{26}}{y^6}
\]
However, from your original options, the closest equivalent expression seems to be:
**64 over x squared y to the 18.** \( \frac{64}{x^2y^{18}} \)
It looks like there might be a misunderstanding or error in the available responses, as the exponential terms and constants do not directly match the fully simplified results we derived.
If you want a direct selection based on your initial options, please recheck the provided options or choose another one based on their relevance, but mathematically, the correct simplification leads to \( \frac{64x^{26}}{y^6} \).
Answer
bot
Answered by
GPT-4o mini
Yes? How can I assist you further? If you have any more questions or need clarification on the previous content, just let me know!
Answer
can u make pics
Answered by
GPT-4o mini
I can't create or display images directly, as I'm text-based. However, I can help you generate descriptions or equations that you could then turn into images using graphic design software or online tools. If you have a specific idea or concept in mind, let me know, and I can assist you with that!
Answer
ok
Apply the product rule of integer exponents to find the numerical equivalent of 12−5⋅127 . Show your work and explain how you used the product rule of integer exponents.(2 points)
Apply the product rule of integer exponents to find the numerical equivalent of 12−5⋅127 . Show your work and explain how you used the product rule of integer exponents.(2 points)
Answered by
GPT-4o mini
To evaluate the expression \(12^{-5} \cdot 12^{7}\) using the product rule of integer exponents, we can follow these steps:
### Step 1: Understand the Product Rule of Integer Exponents
The product rule states that when multiplying two exponents with the same base, you can add the exponents together. This can be expressed as:
\[
a^m \cdot a^n = a^{m+n}
\]
### Step 2: Apply the Product Rule
In this case, we have:
\[
12^{-5} \cdot 12^{7}
\]
According to the product rule:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
### Step 3: Perform the Addition of Exponents
Now, calculate the exponent:
\[
-5 + 7 = 2
\]
So, we have:
\[
12^{-5} \cdot 12^{7} = 12^{2}
\]
### Step 4: Calculate the Result
Now we simplify \(12^{2}\):
\[
12^{2} = 144
\]
### Conclusion
Thus, the numerical equivalent of \(12^{-5} \cdot 12^{7}\) is:
\[
\boxed{144}
\]
### Explanation
In summary, I used the product rule of integer exponents by recognizing that both terms in the multiplication had the same base (12). I added the exponents (-5 and 7) together to find the simplified exponent, resulting in \(12^2\), which I then calculated to find the numerical value of 144.
### Step 1: Understand the Product Rule of Integer Exponents
The product rule states that when multiplying two exponents with the same base, you can add the exponents together. This can be expressed as:
\[
a^m \cdot a^n = a^{m+n}
\]
### Step 2: Apply the Product Rule
In this case, we have:
\[
12^{-5} \cdot 12^{7}
\]
According to the product rule:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
### Step 3: Perform the Addition of Exponents
Now, calculate the exponent:
\[
-5 + 7 = 2
\]
So, we have:
\[
12^{-5} \cdot 12^{7} = 12^{2}
\]
### Step 4: Calculate the Result
Now we simplify \(12^{2}\):
\[
12^{2} = 144
\]
### Conclusion
Thus, the numerical equivalent of \(12^{-5} \cdot 12^{7}\) is:
\[
\boxed{144}
\]
### Explanation
In summary, I used the product rule of integer exponents by recognizing that both terms in the multiplication had the same base (12). I added the exponents (-5 and 7) together to find the simplified exponent, resulting in \(12^2\), which I then calculated to find the numerical value of 144.