Question
Simplify the expression (4x3)2
using the properties of integer exponents.(1 point)
Responses
4x6
4 x superscript 6 baseline
16x6
16 x superscript 6 baseline
16x5
16 x superscript 5 baseline
8x6
using the properties of integer exponents.(1 point)
Responses
4x6
4 x superscript 6 baseline
16x6
16 x superscript 6 baseline
16x5
16 x superscript 5 baseline
8x6
Answers
Answer
Apply the properties of integer exponents to generate an equivalent algebraic expression to (4x4y−4)3(x7y3)2
.(1 point)
Responses
64x2y18
Start Fraction 64 over x squared y superscript 18 baseline End Fraction
x2y1864
Start Fraction x squared y superscript 18 baseline over 64 End Fraction
4x2y18
Start Fraction 4 over x squared y superscript 18 baseline End Fraction
64x20
Start Fraction 64 over x superscript 20 baseline End Fraction
.(1 point)
Responses
64x2y18
Start Fraction 64 over x squared y superscript 18 baseline End Fraction
x2y1864
Start Fraction x squared y superscript 18 baseline over 64 End Fraction
4x2y18
Start Fraction 4 over x squared y superscript 18 baseline End Fraction
64x20
Start Fraction 64 over x superscript 20 baseline End Fraction
Answer
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)
. Show your work and explain how you used the product rule of integer exponents.(2 points)
Answer
use words and make it like a 8 grader wrote
Answer
make it like a 8 grader wrote it
Answer
1 paragraph long
Answered by
GPT-4o mini
To simplify the expression \( (4x^3)^2 \), we can use the properties of exponents. Specifically, we apply two properties:
1. The power of a product: \( (ab)^n = a^n b^n \)
2. The power of a power: \( (a^m)^n = a^{m \cdot n} \)
Using these properties, we have:
\[
(4x^3)^2 = 4^2 (x^3)^2
\]
Calculating each part:
1. \( 4^2 = 16 \)
2. \( (x^3)^2 = x^{3 \cdot 2} = x^6 \)
Putting it all together, we get:
\[
(4x^3)^2 = 16x^6
\]
Thus, the simplified expression is:
**16x\(^6\)**
1. The power of a product: \( (ab)^n = a^n b^n \)
2. The power of a power: \( (a^m)^n = a^{m \cdot n} \)
Using these properties, we have:
\[
(4x^3)^2 = 4^2 (x^3)^2
\]
Calculating each part:
1. \( 4^2 = 16 \)
2. \( (x^3)^2 = x^{3 \cdot 2} = x^6 \)
Putting it all together, we get:
\[
(4x^3)^2 = 16x^6
\]
Thus, the simplified expression is:
**16x\(^6\)**
Answered by
GPT-4o mini
To simplify the expression \( (4x^4y^{-4})^3 (x^7y^3)^2 \), we will apply the properties of exponents step by step.
1. **Apply the power of a product property**:
For \( (4x^4y^{-4})^3 \):
\[
(4)^3 (x^4)^3 (y^{-4})^3 = 64 x^{4 \cdot 3} y^{-4 \cdot 3} = 64 x^{12} y^{-12}
\]
For \( (x^7y^3)^2 \):
\[
(x^7)^2 (y^3)^2 = x^{7 \cdot 2} y^{3 \cdot 2} = x^{14} y^6
\]
2. **Multiply the two results together**:
Now we combine \( (64 x^{12} y^{-12}) \) with \( (x^{14} y^6) \):
\[
64 x^{12} y^{-12} \cdot x^{14} y^6
\]
When multiplying, we add the exponents for like bases:
- For \( x \):
\[
x^{12 + 14} = x^{26}
\]
- For \( y \):
\[
y^{-12 + 6} = y^{-6}
\]
Thus, the expression becomes:
\[
64 x^{26} y^{-6}
\]
3. **Rewrite using positive exponents**:
We can also rewrite \( y^{-6} \) as \( \frac{1}{y^6} \):
\[
64 x^{26} \cdot \frac{1}{y^6} = \frac{64 x^{26}}{y^6}
\]
Now, evaluating the response options:
- **64x²y¹⁸**
- **x²y¹⁸64**
- **4x²y¹⁸**
- **64/x²y¹⁸**
None of the response options exactly match \( \frac{64 x^{26}}{y^6} \).
However, if we analyze it, we have \(64 x^{26} y^{-6}\) or \( \frac{64 x^{26}}{y^6}\).
To finalize, the answer that resembles our final expression the most would be:
**Start Fraction 64 x superscript 26 over y superscript 6 End Fraction (though not listed in the options given).**
If you are looking for a response among the listed options, they are all mismatched with our derived expression. If you're expected to use only the provided responses, please double-check which ones are most similar or consult your instructor.
1. **Apply the power of a product property**:
For \( (4x^4y^{-4})^3 \):
\[
(4)^3 (x^4)^3 (y^{-4})^3 = 64 x^{4 \cdot 3} y^{-4 \cdot 3} = 64 x^{12} y^{-12}
\]
For \( (x^7y^3)^2 \):
\[
(x^7)^2 (y^3)^2 = x^{7 \cdot 2} y^{3 \cdot 2} = x^{14} y^6
\]
2. **Multiply the two results together**:
Now we combine \( (64 x^{12} y^{-12}) \) with \( (x^{14} y^6) \):
\[
64 x^{12} y^{-12} \cdot x^{14} y^6
\]
When multiplying, we add the exponents for like bases:
- For \( x \):
\[
x^{12 + 14} = x^{26}
\]
- For \( y \):
\[
y^{-12 + 6} = y^{-6}
\]
Thus, the expression becomes:
\[
64 x^{26} y^{-6}
\]
3. **Rewrite using positive exponents**:
We can also rewrite \( y^{-6} \) as \( \frac{1}{y^6} \):
\[
64 x^{26} \cdot \frac{1}{y^6} = \frac{64 x^{26}}{y^6}
\]
Now, evaluating the response options:
- **64x²y¹⁸**
- **x²y¹⁸64**
- **4x²y¹⁸**
- **64/x²y¹⁸**
None of the response options exactly match \( \frac{64 x^{26}}{y^6} \).
However, if we analyze it, we have \(64 x^{26} y^{-6}\) or \( \frac{64 x^{26}}{y^6}\).
To finalize, the answer that resembles our final expression the most would be:
**Start Fraction 64 x superscript 26 over y superscript 6 End Fraction (though not listed in the options given).**
If you are looking for a response among the listed options, they are all mismatched with our derived expression. If you're expected to use only the provided responses, please double-check which ones are most similar or consult your instructor.
Answered by
GPT-4o mini
To simplify the expression \( 12^{-5} \cdot 12^{7} \) using the product rule of integer exponents, we apply the following property:
**Product Rule of Exponents**: When multiplying two expressions with the same base, the exponents add together. This property is expressed as:
\[
a^m \cdot a^n = a^{m+n}
\]
### Step 1: Identify the base and exponents.
In this expression:
- Base: \( 12 \)
- Exponent 1: \( -5 \)
- Exponent 2: \( 7 \)
### Step 2: Apply the product rule.
Using the product rule, we can add the exponents:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
### Step 3: Simplify the exponent.
Calculate \( -5 + 7 \):
\[
-5 + 7 = 2
\]
So we have:
\[
12^{-5} \cdot 12^{7} = 12^{2}
\]
### Step 4: Find the numerical equivalent of \( 12^{2} \).
Now, calculate \( 12^{2} \):
\[
12^{2} = 144
\]
### Final Result
Thus, the numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is:
\[
\boxed{144}
\]
### Summary of Steps:
1. Recognized the base (12) and the exponents (-5 and 7).
2. Applied the product rule of exponents: \( a^m \cdot a^n = a^{m+n} \).
3. Simplified the exponent to get \( 12^{2} \).
4. Calculated \( 12^{2} = 144 \).
**Product Rule of Exponents**: When multiplying two expressions with the same base, the exponents add together. This property is expressed as:
\[
a^m \cdot a^n = a^{m+n}
\]
### Step 1: Identify the base and exponents.
In this expression:
- Base: \( 12 \)
- Exponent 1: \( -5 \)
- Exponent 2: \( 7 \)
### Step 2: Apply the product rule.
Using the product rule, we can add the exponents:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
### Step 3: Simplify the exponent.
Calculate \( -5 + 7 \):
\[
-5 + 7 = 2
\]
So we have:
\[
12^{-5} \cdot 12^{7} = 12^{2}
\]
### Step 4: Find the numerical equivalent of \( 12^{2} \).
Now, calculate \( 12^{2} \):
\[
12^{2} = 144
\]
### Final Result
Thus, the numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is:
\[
\boxed{144}
\]
### Summary of Steps:
1. Recognized the base (12) and the exponents (-5 and 7).
2. Applied the product rule of exponents: \( a^m \cdot a^n = a^{m+n} \).
3. Simplified the exponent to get \( 12^{2} \).
4. Calculated \( 12^{2} = 144 \).
Answered by
GPT-4o mini
To solve the expression \( 12^{-5} \cdot 12^{7} \), we can use the product rule of exponents, which helps us combine the two parts that have the same base.
### Step 1: Identify the base and exponents.
In our expression, the base is \( 12 \), and we have two exponents: one is \( -5 \) and the other is \( 7 \).
### Step 2: Apply the product rule.
The product rule of exponents says that when you multiply two numbers that have the same base, you can add their exponents together. This rule looks like this:
\[
a^m \cdot a^n = a^{m+n}
\]
So, for our problem, we can rewrite it by adding the exponents:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
### Step 3: Simplify the exponent.
Now, we just need to add the exponents \(-5\) and \(7\):
\[
-5 + 7 = 2
\]
This means we can simplify our expression to:
\[
12^{2}
\]
### Step 4: Calculate the value.
Finally, we find what \( 12^{2} \) equals. Squaring a number means multiplying it by itself:
\[
12^{2} = 12 \times 12 = 144
\]
### Conclusion
So, the final answer for \( 12^{-5} \cdot 12^{7} \) is \( 144 \). We used the product rule of exponents to combine the exponents, and then we calculated to find the numerical value.
Thus, the answer is:
\[
\boxed{144}
\]
### Step 1: Identify the base and exponents.
In our expression, the base is \( 12 \), and we have two exponents: one is \( -5 \) and the other is \( 7 \).
### Step 2: Apply the product rule.
The product rule of exponents says that when you multiply two numbers that have the same base, you can add their exponents together. This rule looks like this:
\[
a^m \cdot a^n = a^{m+n}
\]
So, for our problem, we can rewrite it by adding the exponents:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
### Step 3: Simplify the exponent.
Now, we just need to add the exponents \(-5\) and \(7\):
\[
-5 + 7 = 2
\]
This means we can simplify our expression to:
\[
12^{2}
\]
### Step 4: Calculate the value.
Finally, we find what \( 12^{2} \) equals. Squaring a number means multiplying it by itself:
\[
12^{2} = 12 \times 12 = 144
\]
### Conclusion
So, the final answer for \( 12^{-5} \cdot 12^{7} \) is \( 144 \). We used the product rule of exponents to combine the exponents, and then we calculated to find the numerical value.
Thus, the answer is:
\[
\boxed{144}
\]
Answered by
GPT-4o mini
To solve the expression \( 12^{-5} \cdot 12^{7} \), I’m going to use the product rule of exponents, which is a handy tool for combining numbers with the same base.
### Step 1: Identify the base and exponents.
First, I look at my expression. The base is \( 12 \), and I see two exponents: one is \( -5 \) and the other is \( 7 \).
### Step 2: Apply the product rule.
The product rule of exponents tells us that if we have the same base, we can just add the exponents. So it looks like this:
\[
a^m \cdot a^n = a^{m + n}
\]
For my expression, that means:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
### Step 3: Simplify the exponent.
Now, I just need to do the math for the exponents:
\[
-5 + 7 = 2
\]
So now my expression is simplified to:
\[
12^{2}
\]
### Step 4: Calculate the value.
Next, I need to figure out what \( 12^{2} \) is. When you square a number, it just means you multiply it by itself:
\[
12^{2} = 12 \times 12 = 144
\]
### Conclusion
So, when I put it all together, the final answer for \( 12^{-5} \cdot 12^{7} \) is \( 144 \). I used the product rule to combine the exponents and then calculated the value.
So, the answer is:
\[
\boxed{144}
\]
### Step 1: Identify the base and exponents.
First, I look at my expression. The base is \( 12 \), and I see two exponents: one is \( -5 \) and the other is \( 7 \).
### Step 2: Apply the product rule.
The product rule of exponents tells us that if we have the same base, we can just add the exponents. So it looks like this:
\[
a^m \cdot a^n = a^{m + n}
\]
For my expression, that means:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
### Step 3: Simplify the exponent.
Now, I just need to do the math for the exponents:
\[
-5 + 7 = 2
\]
So now my expression is simplified to:
\[
12^{2}
\]
### Step 4: Calculate the value.
Next, I need to figure out what \( 12^{2} \) is. When you square a number, it just means you multiply it by itself:
\[
12^{2} = 12 \times 12 = 144
\]
### Conclusion
So, when I put it all together, the final answer for \( 12^{-5} \cdot 12^{7} \) is \( 144 \). I used the product rule to combine the exponents and then calculated the value.
So, the answer is:
\[
\boxed{144}
\]
Answered by
GPT-4o mini
To solve the expression \( 12^{-5} \cdot 12^{7} \), I used the product rule of exponents, which says that when multiplying two expressions with the same base, we can add the exponents. In this case, the base is \( 12 \), and the exponents are \( -5 \) and \( 7 \). So, I combined them by adding: \( -5 + 7 = 2 \). This means the expression simplifies to \( 12^{2} \). Finally, I calculated \( 12^{2} \) by multiplying \( 12 \) by itself, which equals \( 144 \). So, the final answer is \( 144 \).
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