Question
1. Use the Product Rule of Exponents to simplify the expression 2^4 ⋅2^2%0D%0A2%0D%0A4%0D%0A %0D%0A⋅%0D%0A2%0D%0A2%0D%0A .(1 point)
2. Use the Product Rule of Exponents to simplify the expression 3^8 ⋅3^3%0D%0A3%0D%0A8%0D%0A %0D%0A⋅%0D%0A3%0D%0A3%0D%0A.(1 point)
3. Use the Product Rule of Exponents to simplify the expression 7^5 ⋅7^2%0D%0A7%0D%0A5%0D%0A %0D%0A⋅%0D%0A7%0D%0A2%0D%0A .(1 point)
4. Find the numerical equivalent of 7^2⋅7^3%0D%0A7%0D%0A2%0D%0A⋅%0D%0A7%0D%0A3%0D%0A(1 point)
5. Find the numerical equivalent of 3^9⋅3^−5%0D%0A3%0D%0A9%0D%0A⋅%0D%0A3%0D%0A−%0D%0A5%0D%0A(1 point)
2. Use the Product Rule of Exponents to simplify the expression 3^8 ⋅3^3%0D%0A3%0D%0A8%0D%0A %0D%0A⋅%0D%0A3%0D%0A3%0D%0A.(1 point)
3. Use the Product Rule of Exponents to simplify the expression 7^5 ⋅7^2%0D%0A7%0D%0A5%0D%0A %0D%0A⋅%0D%0A7%0D%0A2%0D%0A .(1 point)
4. Find the numerical equivalent of 7^2⋅7^3%0D%0A7%0D%0A2%0D%0A⋅%0D%0A7%0D%0A3%0D%0A(1 point)
5. Find the numerical equivalent of 3^9⋅3^−5%0D%0A3%0D%0A9%0D%0A⋅%0D%0A3%0D%0A−%0D%0A5%0D%0A(1 point)
Answers
GPT-4o mini
Let's simplify each expression step by step using the Product Rule of Exponents, which states that when multiplying two powers with the same base, you add the exponents.
1. **Simplifying \(2^4 \cdot 2^2\)**:
\[
2^4 \cdot 2^2 = 2^{4+2} = 2^6
\]
2. **Simplifying \(3^8 \cdot 3^3\)**:
\[
3^8 \cdot 3^3 = 3^{8+3} = 3^{11}
\]
3. **Simplifying \(7^5 \cdot 7^2\)**:
\[
7^5 \cdot 7^2 = 7^{5+2} = 7^7
\]
4. **Finding the numerical equivalent of \(7^2 \cdot 7^3\)**:
\[
7^2 \cdot 7^3 = 7^{2+3} = 7^5
\]
Now, calculate \(7^5\):
\[
7 \times 7 = 49 \\
49 \times 7 = 343 \\
343 \times 7 = 2401 \\
2401 \times 7 = 16807
\]
Therefore, \(7^5 = 16807\).
5. **Finding the numerical equivalent of \(3^9 \cdot 3^{-5}\)**:
\[
3^9 \cdot 3^{-5} = 3^{9 + (-5)} = 3^{9 - 5} = 3^4
\]
Now, calculate \(3^4\):
\[
3 \times 3 = 9 \\
9 \times 3 = 27 \\
27 \times 3 = 81
\]
Therefore, \(3^4 = 81\).
Here are the final answers:
1. \(2^6\)
2. \(3^{11}\)
3. \(7^7\)
4. \(16807\)
5. \(81\)
1. **Simplifying \(2^4 \cdot 2^2\)**:
\[
2^4 \cdot 2^2 = 2^{4+2} = 2^6
\]
2. **Simplifying \(3^8 \cdot 3^3\)**:
\[
3^8 \cdot 3^3 = 3^{8+3} = 3^{11}
\]
3. **Simplifying \(7^5 \cdot 7^2\)**:
\[
7^5 \cdot 7^2 = 7^{5+2} = 7^7
\]
4. **Finding the numerical equivalent of \(7^2 \cdot 7^3\)**:
\[
7^2 \cdot 7^3 = 7^{2+3} = 7^5
\]
Now, calculate \(7^5\):
\[
7 \times 7 = 49 \\
49 \times 7 = 343 \\
343 \times 7 = 2401 \\
2401 \times 7 = 16807
\]
Therefore, \(7^5 = 16807\).
5. **Finding the numerical equivalent of \(3^9 \cdot 3^{-5}\)**:
\[
3^9 \cdot 3^{-5} = 3^{9 + (-5)} = 3^{9 - 5} = 3^4
\]
Now, calculate \(3^4\):
\[
3 \times 3 = 9 \\
9 \times 3 = 27 \\
27 \times 3 = 81
\]
Therefore, \(3^4 = 81\).
Here are the final answers:
1. \(2^6\)
2. \(3^{11}\)
3. \(7^7\)
4. \(16807\)
5. \(81\)
Im not gonna lie i have no idea how to solve the last two questions..
1. Which expression is correctly developed to use the Product Rule of Exponents?(1 point)%0D%0AResponses%0D%0A%0D%0A62⋅73%0D%0A6 squared times 7 cubed%0D%0A%0D%0A(52)9%0D%0Aleft parenthesis 5 squared right parenthesis superscript 9 baseline%0D%0A%0D%0A327%0D%0A32 superscript 7 baseline%0D%0A%0D%0A108⋅108%0D%0A10 superscript 8 baseline times 10 superscript 8 baseline
2. Which of the following correctly used the Product Rule of Exponents to simplify the expression 1008⋅1007?(1 point)%0D%0AResponses%0D%0A%0D%0A10056%0D%0A100 superscript 56 baseline%0D%0A%0D%0A20015%0D%0A200 superscript 15 baseline%0D%0A%0D%0A10,00015%0D%0A10,000 superscript 15 baseline%0D%0A%0D%0A10015
3. Find the numerical equivalent of 102 ⋅103(1 point)%0D%0AResponses%0D%0A%0D%0A100,000%0D%0A100,000%0D%0A%0D%0A10%0D%0A10%0D%0A%0D%0A1,100%0D%0A1,100%0D%0A%0D%0A105
4. Find the numerical equivalent of 25 ⋅2−2(1 point)%0D%0AResponses%0D%0A%0D%0A16%0D%0A16%0D%0A%0D%0A2%0D%0A2%0D%0A%0D%0A128%0D%0A128%0D%0A%0D%0A8
5. Find the numerical equivalent of 49⋅4−7(1 point)%0D%0AResponses%0D%0A%0D%0A42%0D%0A4 squared%0D%0A%0D%0A4%0D%0A4%0D%0A%0D%0A16%0D%0A16%0D%0A%0D%0A416
1. Which expression is correctly developed to use the Product Rule of Exponents?(1 point)%0D%0AResponses%0D%0A%0D%0A62⋅73%0D%0A6 squared times 7 cubed%0D%0A%0D%0A(52)9%0D%0Aleft parenthesis 5 squared right parenthesis superscript 9 baseline%0D%0A%0D%0A327%0D%0A32 superscript 7 baseline%0D%0A%0D%0A108⋅108%0D%0A10 superscript 8 baseline times 10 superscript 8 baseline
2. Which of the following correctly used the Product Rule of Exponents to simplify the expression 1008⋅1007?(1 point)%0D%0AResponses%0D%0A%0D%0A10056%0D%0A100 superscript 56 baseline%0D%0A%0D%0A20015%0D%0A200 superscript 15 baseline%0D%0A%0D%0A10,00015%0D%0A10,000 superscript 15 baseline%0D%0A%0D%0A10015
3. Find the numerical equivalent of 102 ⋅103(1 point)%0D%0AResponses%0D%0A%0D%0A100,000%0D%0A100,000%0D%0A%0D%0A10%0D%0A10%0D%0A%0D%0A1,100%0D%0A1,100%0D%0A%0D%0A105
4. Find the numerical equivalent of 25 ⋅2−2(1 point)%0D%0AResponses%0D%0A%0D%0A16%0D%0A16%0D%0A%0D%0A2%0D%0A2%0D%0A%0D%0A128%0D%0A128%0D%0A%0D%0A8
5. Find the numerical equivalent of 49⋅4−7(1 point)%0D%0AResponses%0D%0A%0D%0A42%0D%0A4 squared%0D%0A%0D%0A4%0D%0A4%0D%0A%0D%0A16%0D%0A16%0D%0A%0D%0A416
GPT-4o mini
Let's go through each question step by step.
### Question 1: Which expression is correctly developed to use the Product Rule of Exponents?
The Product Rule of Exponents states that when multiplying two powers with the same base, the exponents are added. The expression that fits this rule needs to have terms multiplied that share the same base.
1. **\(6^2 \cdot 7^3\)** - This represents multiplication of bases \(6\) and \(7\), not applicable.
2. **\((5^2)^9\)** - This uses the power of a power rule, not the product rule.
3. **\(3^2 \cdot 7\)** - \(3^2\) is being multiplied by \(7\), not applicable.
4. **\(10^8 \cdot 10^8\)** - This has the same base \(10\), and since it can be simplified using the product rule:
\[
10^8 \cdot 10^8 = 10^{8+8} = 10^{16}
\]
This is the correct option.
**Answer: \(10^8 \cdot 10^8\)**
### Question 2: Which of the following correctly used the Product Rule of Exponents to simplify the expression \(10^8 \cdot 10^7\)?
Using the Product Rule:
\[
10^8 \cdot 10^7 = 10^{8+7} = 10^{15}
\]
1. **\(10^{56}\)** - Incorrect.
2. **\(20^{15}\)** - Incorrect.
3. **\(10,000^{15}\)** - Incorrect.
4. **\(10^{15}\)** - Correct.
**Answer: \(10^{15}\)**
### Question 3: Find the numerical equivalent of \(10^2 \cdot 10^3\).
Using the Product Rule:
\[
10^2 \cdot 10^3 = 10^{2+3} = 10^5
\]
Now, calculate \(10^5\):
\[
10^5 = 100,000
\]
**Answer: \(100,000\)**
### Question 4: Find the numerical equivalent of \(2^5 \cdot 2^{-2}\).
Using the Product Rule:
\[
2^5 \cdot 2^{-2} = 2^{5 + (-2)} = 2^{5 - 2} = 2^3
\]
Now, calculate \(2^3\):
\[
2^3 = 8
\]
**Answer: \(8\)**
### Question 5: Find the numerical equivalent of \(4^9 \cdot 4^{-7}\).
Using the Product Rule:
\[
4^9 \cdot 4^{-7} = 4^{9 + (-7)} = 4^{9 - 7} = 4^2
\]
Now, calculate \(4^2\):
\[
4^2 = 16
\]
**Answer: \(16\)**
Here are the final answers summarized:
1. \(10^8 \cdot 10^8\)
2. \(10^{15}\)
3. \(100,000\)
4. \(8\)
5. \(16\)
### Question 1: Which expression is correctly developed to use the Product Rule of Exponents?
The Product Rule of Exponents states that when multiplying two powers with the same base, the exponents are added. The expression that fits this rule needs to have terms multiplied that share the same base.
1. **\(6^2 \cdot 7^3\)** - This represents multiplication of bases \(6\) and \(7\), not applicable.
2. **\((5^2)^9\)** - This uses the power of a power rule, not the product rule.
3. **\(3^2 \cdot 7\)** - \(3^2\) is being multiplied by \(7\), not applicable.
4. **\(10^8 \cdot 10^8\)** - This has the same base \(10\), and since it can be simplified using the product rule:
\[
10^8 \cdot 10^8 = 10^{8+8} = 10^{16}
\]
This is the correct option.
**Answer: \(10^8 \cdot 10^8\)**
### Question 2: Which of the following correctly used the Product Rule of Exponents to simplify the expression \(10^8 \cdot 10^7\)?
Using the Product Rule:
\[
10^8 \cdot 10^7 = 10^{8+7} = 10^{15}
\]
1. **\(10^{56}\)** - Incorrect.
2. **\(20^{15}\)** - Incorrect.
3. **\(10,000^{15}\)** - Incorrect.
4. **\(10^{15}\)** - Correct.
**Answer: \(10^{15}\)**
### Question 3: Find the numerical equivalent of \(10^2 \cdot 10^3\).
Using the Product Rule:
\[
10^2 \cdot 10^3 = 10^{2+3} = 10^5
\]
Now, calculate \(10^5\):
\[
10^5 = 100,000
\]
**Answer: \(100,000\)**
### Question 4: Find the numerical equivalent of \(2^5 \cdot 2^{-2}\).
Using the Product Rule:
\[
2^5 \cdot 2^{-2} = 2^{5 + (-2)} = 2^{5 - 2} = 2^3
\]
Now, calculate \(2^3\):
\[
2^3 = 8
\]
**Answer: \(8\)**
### Question 5: Find the numerical equivalent of \(4^9 \cdot 4^{-7}\).
Using the Product Rule:
\[
4^9 \cdot 4^{-7} = 4^{9 + (-7)} = 4^{9 - 7} = 4^2
\]
Now, calculate \(4^2\):
\[
4^2 = 16
\]
**Answer: \(16\)**
Here are the final answers summarized:
1. \(10^8 \cdot 10^8\)
2. \(10^{15}\)
3. \(100,000\)
4. \(8\)
5. \(16\)