7 Multiple Choice 1 point The least common denominator for $$ \frac{2}{12 x^{4} y^{3}}-\frac{1}{3 x^{2} y^{5}} $$ is $$ 3 x^{2} y^{3} $$ $$ 36 x^{4} y^{5} $$ $$ 12 x^{4} y^{5} $$ $$ 36 x^{6} y^{8} $$

Question

UpStudy AI · Accepted Answer

To find the least common denominator (LCD) for the given fractions, we first need to identify the denominators:

and

.

We start with the numerical coefficients: the least common multiple (LCM) of

and

is

.

Next, we consider the variable parts. For

, the highest power is

, and for

, the highest power is

.

Putting this all together, the least common denominator is

.

So the correct answer is

.

1. **Factor the denominators:**
   - $$ 12 = 2^2 \cdot 3 $$
   - $$ 3 = 3^1 $$

So, the first denominator can be factored as:
   $$
   12 x^{4} y^{3} = 2^2 \cdot 3^1 \cdot x^{4} \cdot y^{3}
   $$

The second denominator can be factored as:
   $$
   3 x^{2} y^{5} = 3^1 \cdot x^{2} \cdot y^{5}
   $$

2. **Find the LCM of the coefficients:**
   - The LCM of $$ 12 $$ and $$ 3 $$ is $$ 12 $$ (since $$ 12 $$ is the highest power of $$ 2 $$ and $$ 3 $$ present).

3. **Find the LCM of the variable parts:**
   - For $$ x^{4} $$ and $$ x^{2} $$, the LCM is $$ x^{4} $$ (the highest power).
   - For $$ y^{3} $$ and $$ y^{5} $$, the LCM is $$ y^{5} $$ (the highest power).

4. **Combine the LCMs:**
   $$
   \text{LCD} = 12 \cdot x^{4} \cdot y^{5} = 12 x^{4} y^{5}
   $$

Thus, the least common denominator for the given fractions is $$ 12 x^{4} y^{5} $$.

The correct answer is:
$$ 12 x^{4} y^{5} $$
The least common denominator is $$ 12 x^{4} y^{5} $$.

7 Multiple Choice 1 point
The least common denominator for is

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7 Multiple Choice 1 point The least common denominator for is