Question
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)
Math Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is correct. The student made no errors while solving.
The answer shows that the student used problem-solving skills and reasoning.
All work is shown or explained.
1
The answer is partially correct. The student made errors while solving.
The answer shows little problem-solving skills or reasoning.
Some work is shown or explained.
0 The question is not answered.
Math Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is correct. The student made no errors while solving.
The answer shows that the student used problem-solving skills and reasoning.
All work is shown or explained.
1
The answer is partially correct. The student made errors while solving.
The answer shows little problem-solving skills or reasoning.
Some work is shown or explained.
0 The question is not answered.
Answers
GPT-4o mini
To solve the expression \( 12^{-5} \cdot 12^7 \) using the product rule of integer exponents, we start by recalling the product rule itself, which states:
\[
a^m \cdot a^n = a^{m+n}
\]
where \( a \) is the base and \( m \) and \( n \) are the exponents.
### Step 1: Identify the base and exponents
In our expression, the base is \( 12 \), and we have two exponents: \( -5 \) and \( 7 \).
### Step 2: Apply the product rule
Using the product rule, we can add the exponents of the same base:
\[
12^{-5} \cdot 12^{7} = 12^{-5+7}
\]
### Step 3: Simplify the exponent
Calculate the exponent:
\[
-5 + 7 = 2
\]
### Step 4: Rewrite the expression
Now we can rewrite the expression using the simplified exponent:
\[
12^{2}
\]
### Step 5: Calculate \( 12^{2} \)
Finally, we calculate \( 12^{2} \):
\[
12^{2} = 144
\]
### Conclusion
The numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is \( 144 \).
### Explanation of the process
I used the product rule of integer exponents effectively by identifying the common base (12), applying the rule to combine the exponents, and then calculating the resulting exponent to find the final numerical value.
\[
a^m \cdot a^n = a^{m+n}
\]
where \( a \) is the base and \( m \) and \( n \) are the exponents.
### Step 1: Identify the base and exponents
In our expression, the base is \( 12 \), and we have two exponents: \( -5 \) and \( 7 \).
### Step 2: Apply the product rule
Using the product rule, we can add the exponents of the same base:
\[
12^{-5} \cdot 12^{7} = 12^{-5+7}
\]
### Step 3: Simplify the exponent
Calculate the exponent:
\[
-5 + 7 = 2
\]
### Step 4: Rewrite the expression
Now we can rewrite the expression using the simplified exponent:
\[
12^{2}
\]
### Step 5: Calculate \( 12^{2} \)
Finally, we calculate \( 12^{2} \):
\[
12^{2} = 144
\]
### Conclusion
The numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is \( 144 \).
### Explanation of the process
I used the product rule of integer exponents effectively by identifying the common base (12), applying the rule to combine the exponents, and then calculating the resulting exponent to find the final numerical value.