Apply the properties of integer exponents to generate an equivalent algebraic expression to 3x^3y^-4/12x^y^4

Question

3/12x^4y8

1/4x^4y8

1/4x^12

4x^4y^8

Answer 1

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 \( \frac{x^2}{4y^8} \).

Answer 2

Which one

Answer 3

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.

Answer 4

Apply the properties of integer exponents to simplify the algebraic expression of (2x6^3 * (x^2y)^3

Answer 5

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 \( 2x6^3 \cdot (x^2y)^3 \).

Answer 6

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

Answer 7

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 \( 2x6^3 \cdot (x^2y)^3 \).