Question
Apply the Properties of Integer Exponents to generate equivalent expressions to 3^7⋅3^−9%0D%0A3%0D%0A7%0D%0A⋅%0D%0A3%0D%0A−%0D%0A9%0D%0A with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
ply the Properties of Integer Exponents to generate equivalent expressions to 2^2/2^−4%0D%0A2%0D%0A2%0D%0A2%0D%0A−%0D%0A4%0D%0A with only positive exponents, then solve the expression. Provide the solution as a whole number.(1 point)%0D%0A222−4=%0D%0A2%0D%0A2%0D%0A2%0D%0A−
%0D%0A4%0D%0A=%0D%0A
3. Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 x 5^−2/(−2)^−3%0D%0A4%0D%0A0%0D%0A %0D%0A⋅%0D%0A %0D%0A5%0D%0A−%0D%0A2%0D%0A(%0D%0A−%0D%0A2%0D%0A)%0D%0A−%0D%0A3%0D%0A with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
4. Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3x27^−7)^0/3^4%0D%0A(%0D%0A14%0D%0A3%0D%0A⋅%0D%0A27%0D%0A−%0D%0A7%0D%0A)%0D%0A0%0D%0A3%0D%0A4%0D%0A with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13%0D%0A(%0D%0A4%0D%0A3%0D%0A %0D%0A⋅%0D%0A %0D%0A7%0D%0A−%0D%0A2%0D%0A)%0D%0A−%0D%0A5%0D%0A7%0D%0A9%0D%0A %0D%0A⋅%0D%0A %0D%0A4%0D%0A−%0D%0A13%0D%0A with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)%0D%0A(4^3x7^−2)^−5/7^9x4^−13=%0D%0A(%0D%0A4%0D%0A3%0D%0A %0D%0A⋅%0D%0A %0D%0A7%0D%0A−%0D%0A2%0D%0A)%0D%0A−%0D%0A5%0D%0A7%0D%0A9%0D%0A %0D%0A⋅%0D%0A %0D%0A4%0D%0A−%0D%0A13%0D%0A=%0D%0A
ply the Properties of Integer Exponents to generate equivalent expressions to 2^2/2^−4%0D%0A2%0D%0A2%0D%0A2%0D%0A−%0D%0A4%0D%0A with only positive exponents, then solve the expression. Provide the solution as a whole number.(1 point)%0D%0A222−4=%0D%0A2%0D%0A2%0D%0A2%0D%0A−
%0D%0A4%0D%0A=%0D%0A
3. Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 x 5^−2/(−2)^−3%0D%0A4%0D%0A0%0D%0A %0D%0A⋅%0D%0A %0D%0A5%0D%0A−%0D%0A2%0D%0A(%0D%0A−%0D%0A2%0D%0A)%0D%0A−%0D%0A3%0D%0A with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
4. Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3x27^−7)^0/3^4%0D%0A(%0D%0A14%0D%0A3%0D%0A⋅%0D%0A27%0D%0A−%0D%0A7%0D%0A)%0D%0A0%0D%0A3%0D%0A4%0D%0A with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13%0D%0A(%0D%0A4%0D%0A3%0D%0A %0D%0A⋅%0D%0A %0D%0A7%0D%0A−%0D%0A2%0D%0A)%0D%0A−%0D%0A5%0D%0A7%0D%0A9%0D%0A %0D%0A⋅%0D%0A %0D%0A4%0D%0A−%0D%0A13%0D%0A with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)%0D%0A(4^3x7^−2)^−5/7^9x4^−13=%0D%0A(%0D%0A4%0D%0A3%0D%0A %0D%0A⋅%0D%0A %0D%0A7%0D%0A−%0D%0A2%0D%0A)%0D%0A−%0D%0A5%0D%0A7%0D%0A9%0D%0A %0D%0A⋅%0D%0A %0D%0A4%0D%0A−%0D%0A13%0D%0A=%0D%0A
Answers
Answer
heres the quick check as well!! 1. 1/5^4
2. the Property of Negative Exponents and the Product Rule of Exponents
3. (-4)^3/7^2
4. 64
5. 2^3/9^2
2. the Property of Negative Exponents and the Product Rule of Exponents
3. (-4)^3/7^2
4. 64
5. 2^3/9^2
Answered by
GPT-4o mini
Let's solve each expression step by step using the properties of integer exponents.
### 1. Evaluate \( 3^7 \cdot 3^{-9} \)
Using the product of powers property \( a^m \cdot a^n = a^{m+n} \):
\[
3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{-2}
\]
Now converting to positive exponents:
\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\]
**Final Answer**: \(\frac{1}{9}\)
---
### 2. Evaluate \( \frac{2^2}{2^{-4}} \)
Using the quotient of powers property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{2^2}{2^{-4}} = 2^{2 - (-4)} = 2^{2 + 4} = 2^6
\]
Calculating \( 2^6 \):
\[
2^6 = 64
\]
**Final Answer**: \(64\)
---
### 3. Evaluate \( \frac{4^0 \cdot 5^{-2}}{(-2)^{-3}} \)
First, simplify each term:
- \( 4^0 = 1 \)
- \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \)
- \( (-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8} \)
Now plug it back into the expression:
\[
\frac{1 \cdot \frac{1}{25}}{-\frac{1}{8}} = \frac{\frac{1}{25}}{-\frac{1}{8}} = \frac{1}{25} \cdot -8 = -\frac{8}{25}
\]
**Final Answer**: \(-\frac{8}{25}\)
---
### 4. Evaluate \( \frac{(14^3 \cdot 27^{-7})^0}{3^4} \)
Using the property that anything raised to the power of zero is 1:
\[
(14^3 \cdot 27^{-7})^0 = 1
\]
So the expression becomes:
\[
\frac{1}{3^4} = \frac{1}{81}
\]
**Final Answer**: \(\frac{1}{81}\)
---
### 5. Evaluate \( \frac{(4^3 \cdot 7^{-2})^{-5}}{7^9 \cdot 4^{-13}} \)
Using the power of a product property and converting to positive exponents:
\[
(4^3 \cdot 7^{-2})^{-5} = 4^{-15} \cdot 7^{10}
\]
Now rewrite the original expression:
\[
\frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} = \frac{4^{-15}}{4^{-13}} \cdot \frac{7^{10}}{7^9}
\]
By applying the properties:
\[
\frac{4^{-15}}{4^{-13}} = 4^{-15 - (-13)} = 4^{-2} = \frac{1}{4^2} = \frac{1}{16}
\]
\[
\frac{7^{10}}{7^9} = 7^{10 - 9} = 7^{1} = 7
\]
Combining the results:
\[
\frac{1}{16} \cdot 7 = \frac{7}{16}
\]
**Final Answer**: \(\frac{7}{16}\)
---
This concludes the evaluations with equivalent expressions and simplified fractions.
### 1. Evaluate \( 3^7 \cdot 3^{-9} \)
Using the product of powers property \( a^m \cdot a^n = a^{m+n} \):
\[
3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{-2}
\]
Now converting to positive exponents:
\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\]
**Final Answer**: \(\frac{1}{9}\)
---
### 2. Evaluate \( \frac{2^2}{2^{-4}} \)
Using the quotient of powers property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{2^2}{2^{-4}} = 2^{2 - (-4)} = 2^{2 + 4} = 2^6
\]
Calculating \( 2^6 \):
\[
2^6 = 64
\]
**Final Answer**: \(64\)
---
### 3. Evaluate \( \frac{4^0 \cdot 5^{-2}}{(-2)^{-3}} \)
First, simplify each term:
- \( 4^0 = 1 \)
- \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \)
- \( (-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8} \)
Now plug it back into the expression:
\[
\frac{1 \cdot \frac{1}{25}}{-\frac{1}{8}} = \frac{\frac{1}{25}}{-\frac{1}{8}} = \frac{1}{25} \cdot -8 = -\frac{8}{25}
\]
**Final Answer**: \(-\frac{8}{25}\)
---
### 4. Evaluate \( \frac{(14^3 \cdot 27^{-7})^0}{3^4} \)
Using the property that anything raised to the power of zero is 1:
\[
(14^3 \cdot 27^{-7})^0 = 1
\]
So the expression becomes:
\[
\frac{1}{3^4} = \frac{1}{81}
\]
**Final Answer**: \(\frac{1}{81}\)
---
### 5. Evaluate \( \frac{(4^3 \cdot 7^{-2})^{-5}}{7^9 \cdot 4^{-13}} \)
Using the power of a product property and converting to positive exponents:
\[
(4^3 \cdot 7^{-2})^{-5} = 4^{-15} \cdot 7^{10}
\]
Now rewrite the original expression:
\[
\frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} = \frac{4^{-15}}{4^{-13}} \cdot \frac{7^{10}}{7^9}
\]
By applying the properties:
\[
\frac{4^{-15}}{4^{-13}} = 4^{-15 - (-13)} = 4^{-2} = \frac{1}{4^2} = \frac{1}{16}
\]
\[
\frac{7^{10}}{7^9} = 7^{10 - 9} = 7^{1} = 7
\]
Combining the results:
\[
\frac{1}{16} \cdot 7 = \frac{7}{16}
\]
**Final Answer**: \(\frac{7}{16}\)
---
This concludes the evaluations with equivalent expressions and simplified fractions.
Answered by
GPT-4o mini
Thanks for providing the quick check values! Let's verify whether those provided answers and principles coincide with our detailed calculations.
### Quick Check Answers vs. Our Solutions
1. **Your Answer**: \(\frac{1}{5^4}\)
- In the original evaluation for \(3^7 \cdot 3^{-9}\), we ended up with:
\[
\frac{1}{9}
\]
- **Comment**: This appears to be a mix-up. The calculation should still conclude as \(\frac{1}{9}\).
2. **Your Claim**: Uses the Property of Negative Exponents and the Product Rule of Exponents.
- In our evaluation, we used:
\[
\frac{2^2}{2^{-4}} = 2^{2 - (-4)} = 2^6
\]
- This correctly utilized the properties outlined.
3. **Your Answer**: \((-4)^3/7^2\)
- For the fraction \(\frac{4^0 \cdot 5^{-2}}{(-2)^{-3}}\), we found:
\[
-\frac{8}{25}
\]
- **Comment**: It seems you might have intended to alter the parameters here, as this result does not match.
4. **Your Answer**: \(64\)
- The expression \(\frac{(14^3 \cdot 27^{-7})^0}{3^4}\) yields:
\[
\frac{1}{81}
\]
- **Comment**: The result seems divergent from what has been shared.
5. **Your Answer**: \(\frac{2^3}{9^2}\)
- From the evaluation of \(\frac{(4^3 \cdot 7^{-2})^{-5}}{7^9 \cdot 4^{-13}}\), we derived:
\[
\frac{7}{16}
\]
- **Comment**: Again, this expression does not correspond directly to the conclusion we reached.
### Summary
- While I appreciate your input with the quick check answers, there are discrepancies between those answers and the calculations provided earlier.
- For clarity and accuracy in your studies, I suggest revisiting those evaluations or share the context of the quick check to ensure there's understanding on potential differences in interpretations or specific terms utilized. Feel free to ask further questions or for clarification on any of the properties discussed!
### Quick Check Answers vs. Our Solutions
1. **Your Answer**: \(\frac{1}{5^4}\)
- In the original evaluation for \(3^7 \cdot 3^{-9}\), we ended up with:
\[
\frac{1}{9}
\]
- **Comment**: This appears to be a mix-up. The calculation should still conclude as \(\frac{1}{9}\).
2. **Your Claim**: Uses the Property of Negative Exponents and the Product Rule of Exponents.
- In our evaluation, we used:
\[
\frac{2^2}{2^{-4}} = 2^{2 - (-4)} = 2^6
\]
- This correctly utilized the properties outlined.
3. **Your Answer**: \((-4)^3/7^2\)
- For the fraction \(\frac{4^0 \cdot 5^{-2}}{(-2)^{-3}}\), we found:
\[
-\frac{8}{25}
\]
- **Comment**: It seems you might have intended to alter the parameters here, as this result does not match.
4. **Your Answer**: \(64\)
- The expression \(\frac{(14^3 \cdot 27^{-7})^0}{3^4}\) yields:
\[
\frac{1}{81}
\]
- **Comment**: The result seems divergent from what has been shared.
5. **Your Answer**: \(\frac{2^3}{9^2}\)
- From the evaluation of \(\frac{(4^3 \cdot 7^{-2})^{-5}}{7^9 \cdot 4^{-13}}\), we derived:
\[
\frac{7}{16}
\]
- **Comment**: Again, this expression does not correspond directly to the conclusion we reached.
### Summary
- While I appreciate your input with the quick check answers, there are discrepancies between those answers and the calculations provided earlier.
- For clarity and accuracy in your studies, I suggest revisiting those evaluations or share the context of the quick check to ensure there's understanding on potential differences in interpretations or specific terms utilized. Feel free to ask further questions or for clarification on any of the properties discussed!