Asked by Milana
Apply the properties of integer exponents to generate an equivalent algebraic expression to 3x^3y^-4/12x^y^4
3/12x^4y8
1/4x^4y8
1/4x^12
4x^4y^8
3/12x^4y8
1/4x^4y8
1/4x^12
4x^4y^8
Answers
Answered by
Milana
Which one
Answered by
Milana
Apply the properties of integer exponents to simplify the algebraic expression of (2x6^3 * (x^2y)^3
Answered by
Milana
Apply the properties of integer exponents to simplify the algebraic expression of (2x6^3 * (x^2y)^3
16x^25
32x^10y^11
X^10y15
16x^10y15
16x^25
32x^10y^11
X^10y15
16x^10y15
Answered by
GPT-4o mini
To simplify the expression \( \frac{3x^3y^{-4}}{12x y^4} \) using the properties of integer exponents, we can follow these steps:
1. **Separate the coefficients and the variables**:
\[
\frac{3}{12} \cdot \frac{x^3}{x} \cdot \frac{y^{-4}}{y^4}
\]
2. **Simplify the coefficients**:
\[
\frac{3}{12} = \frac{1}{4}
\]
3. **Simplify the \( x \) terms using the property \( \frac{x^a}{x^b} = x^{a-b} \)**:
\[
\frac{x^3}{x} = x^{3-1} = x^2
\]
4. **Simplify the \( y \) terms using the property \( \frac{y^a}{y^b} = y^{a-b} \)**:
\[
\frac{y^{-4}}{y^4} = y^{-4-4} = y^{-8}
\]
5. **Combine everything**:
\[
\frac{1}{4} x^2 y^{-8}
\]
6. **Rearranging to express it with positive exponents**:
\[
\frac{x^2}{4y^8}
\]
Thus, the equivalent algebraic expression to \( \frac{3x^3y^{-4}}{12xy^4} \) is:
\[
\frac{x^2}{4y^8}
\]
None of the provided choices match this exact expression, but let's check which is most similar:
- \( \frac{3}{12x^4y^8} \) simplifies to \( \frac{1}{4x^4y^8} \)
- \( \frac{1}{4x^4y^8} \) does not match.
- \( \frac{1}{4x^{12}} \) does not match.
- \( 4x^4y^8 \) does not match.
Therefore, the closest choice that simplifies correctly would be **\( \frac{1}{4}x^2y^{-8} \)** (if presented in the selection with all positive exponents).
Given the provided options, it seems like the correct consideration is not present, but if sticking strictly to provided expressions, **\( \frac{1}{4x^4y^8} \)** can be marked as the simplest equivalent structure available, yet the primary answer from simplification is \( \frac{x^2}{4y^8} \).
1. **Separate the coefficients and the variables**:
\[
\frac{3}{12} \cdot \frac{x^3}{x} \cdot \frac{y^{-4}}{y^4}
\]
2. **Simplify the coefficients**:
\[
\frac{3}{12} = \frac{1}{4}
\]
3. **Simplify the \( x \) terms using the property \( \frac{x^a}{x^b} = x^{a-b} \)**:
\[
\frac{x^3}{x} = x^{3-1} = x^2
\]
4. **Simplify the \( y \) terms using the property \( \frac{y^a}{y^b} = y^{a-b} \)**:
\[
\frac{y^{-4}}{y^4} = y^{-4-4} = y^{-8}
\]
5. **Combine everything**:
\[
\frac{1}{4} x^2 y^{-8}
\]
6. **Rearranging to express it with positive exponents**:
\[
\frac{x^2}{4y^8}
\]
Thus, the equivalent algebraic expression to \( \frac{3x^3y^{-4}}{12xy^4} \) is:
\[
\frac{x^2}{4y^8}
\]
None of the provided choices match this exact expression, but let's check which is most similar:
- \( \frac{3}{12x^4y^8} \) simplifies to \( \frac{1}{4x^4y^8} \)
- \( \frac{1}{4x^4y^8} \) does not match.
- \( \frac{1}{4x^{12}} \) does not match.
- \( 4x^4y^8 \) does not match.
Therefore, the closest choice that simplifies correctly would be **\( \frac{1}{4}x^2y^{-8} \)** (if presented in the selection with all positive exponents).
Given the provided options, it seems like the correct consideration is not present, but if sticking strictly to provided expressions, **\( \frac{1}{4x^4y^8} \)** can be marked as the simplest equivalent structure available, yet the primary answer from simplification is \( \frac{x^2}{4y^8} \).
Answered by
GPT-4o mini
From the given options:
1. \( \frac{3}{12x^4y^8} \)
2. \( \frac{1}{4x^4y^8} \)
3. \( \frac{1}{4x^{12}} \)
4. \( 4x^4y^8 \)
The expression we simplified earlier is:
\[
\frac{x^2}{4y^8}
\]
This can be reformulated (to match the provided options) as:
\[
\frac{1}{4} \cdot \frac{x^2}{y^8}
\]
However, none of the options directly matches this expression. The option that is closest to being in the form we derived is:
**2. \( \frac{1}{4x^4y^8} \)**
This is the only choice that is in the form of a fraction with \(4\) in the denominator, though it does not fully represent the expression we simplified to. If forced to choose from the provided options, **\( \frac{1}{4x^4y^8} \)** is the best match, but please note that it does not fully represent the simplified form of the initial expression.
1. \( \frac{3}{12x^4y^8} \)
2. \( \frac{1}{4x^4y^8} \)
3. \( \frac{1}{4x^{12}} \)
4. \( 4x^4y^8 \)
The expression we simplified earlier is:
\[
\frac{x^2}{4y^8}
\]
This can be reformulated (to match the provided options) as:
\[
\frac{1}{4} \cdot \frac{x^2}{y^8}
\]
However, none of the options directly matches this expression. The option that is closest to being in the form we derived is:
**2. \( \frac{1}{4x^4y^8} \)**
This is the only choice that is in the form of a fraction with \(4\) in the denominator, though it does not fully represent the expression we simplified to. If forced to choose from the provided options, **\( \frac{1}{4x^4y^8} \)** is the best match, but please note that it does not fully represent the simplified form of the initial expression.
Answered by
GPT-4o mini
To simplify the expression \( 2x6^3 \cdot (x^2y)^3 \), we will apply the properties of integer exponents step by step.
1. **Simplify \( 6^3 \)**:
\[
6^3 = 216
\]
So,
\[
2 \cdot 6^3 = 2 \cdot 216 = 432
\]
2. **Expand \( (x^2y)^3 \)** using the power of a product rule, \( (ab)^m = a^m \cdot b^m \):
\[
(x^2y)^3 = (x^2)^3 \cdot (y)^3 = x^{2 \cdot 3} \cdot y^3 = x^6y^3
\]
3. **Combine the expressions**:
Now substituting back into the expression gives us:
\[
432x \cdot x^6y^3
\]
4. **Combine the \(x\) terms using the property \( x^a \cdot x^b = x^{a+b} \)**:
\[
432x^{1 + 6}y^3 = 432x^7y^3
\]
The fully simplified expression is:
\[
432x^7y^3
\]
This is the simplified version of the given expression \( 2x6^3 \cdot (x^2y)^3 \).
1. **Simplify \( 6^3 \)**:
\[
6^3 = 216
\]
So,
\[
2 \cdot 6^3 = 2 \cdot 216 = 432
\]
2. **Expand \( (x^2y)^3 \)** using the power of a product rule, \( (ab)^m = a^m \cdot b^m \):
\[
(x^2y)^3 = (x^2)^3 \cdot (y)^3 = x^{2 \cdot 3} \cdot y^3 = x^6y^3
\]
3. **Combine the expressions**:
Now substituting back into the expression gives us:
\[
432x \cdot x^6y^3
\]
4. **Combine the \(x\) terms using the property \( x^a \cdot x^b = x^{a+b} \)**:
\[
432x^{1 + 6}y^3 = 432x^7y^3
\]
The fully simplified expression is:
\[
432x^7y^3
\]
This is the simplified version of the given expression \( 2x6^3 \cdot (x^2y)^3 \).
Answered by
GPT-4o mini
Let's simplify the expression \( 2x6^3 \cdot (x^2y)^3 \) step by step using the properties of integer exponents.
### Step 1: Simplify \( 2x6^3 \)
1. Calculate \( 6^3 \):
\[
6^3 = 216
\]
2. Therefore:
\[
2 \cdot 6^3 = 2 \cdot 216 = 432
\]
3. So, we have:
\[
432x
\]
### Step 2: Simplify \( (x^2y)^3 \)
Using the property \( (ab)^m = a^m \cdot b^m \):
\[
(x^2y)^3 = (x^2)^3 \cdot y^3 = x^{2 \cdot 3} \cdot y^3 = x^6 \cdot y^3
\]
### Step 3: Combine the results
Now substitute back into the entire expression:
\[
432x \cdot (x^6y^3) = 432x^{1 + 6}y^3 = 432x^7y^3
\]
### Summary of the Simplification
The expression simplifies to:
\[
432x^7y^3
\]
### Match with Given Options
Now let's compare this with the options you've provided:
- **16x^25**
- **32x^10y^11**
- **x^{10}y^{15}**
- **16x^{10}y^{15}**
None of the options match \( 432x^7y^3 \). There seems to be a discrepancy in the choices provided. If we need to verify any other values, or if the initial problem has variations, please provide that so I can assist you further. Otherwise, given the calculations, \( 432x^7y^3 \) is the correct simplification of the expression \( 2x6^3 \cdot (x^2y)^3 \).
### Step 1: Simplify \( 2x6^3 \)
1. Calculate \( 6^3 \):
\[
6^3 = 216
\]
2. Therefore:
\[
2 \cdot 6^3 = 2 \cdot 216 = 432
\]
3. So, we have:
\[
432x
\]
### Step 2: Simplify \( (x^2y)^3 \)
Using the property \( (ab)^m = a^m \cdot b^m \):
\[
(x^2y)^3 = (x^2)^3 \cdot y^3 = x^{2 \cdot 3} \cdot y^3 = x^6 \cdot y^3
\]
### Step 3: Combine the results
Now substitute back into the entire expression:
\[
432x \cdot (x^6y^3) = 432x^{1 + 6}y^3 = 432x^7y^3
\]
### Summary of the Simplification
The expression simplifies to:
\[
432x^7y^3
\]
### Match with Given Options
Now let's compare this with the options you've provided:
- **16x^25**
- **32x^10y^11**
- **x^{10}y^{15}**
- **16x^{10}y^{15}**
None of the options match \( 432x^7y^3 \). There seems to be a discrepancy in the choices provided. If we need to verify any other values, or if the initial problem has variations, please provide that so I can assist you further. Otherwise, given the calculations, \( 432x^7y^3 \) is the correct simplification of the expression \( 2x6^3 \cdot (x^2y)^3 \).
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