6. $$ \frac{\left(2 x^{3} y^{4}\right)^{5}\left(3 x^{2} y^{7}\right)^{2}}{6 x^{-5} y^{15}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{\left(2 x^{3} y^{4}\right)^{5}\left(3 x^{2} y^{7}\right)^{2}}{6 x^{-5} y^{15}}
$$

Let's break it down step-by-step.

### Step 1: Expand the Numerator

First, apply the exponents to each part inside the parentheses:

$$
(2 x^{3} y^{4})^{5} = 2^{5} \cdot x^{3 \times 5} \cdot y^{4 \times 5} = 32 x^{15} y^{20}
$$

$$
(3 x^{2} y^{7})^{2} = 3^{2} \cdot x^{2 \times 2} \cdot y^{7 \times 2} = 9 x^{4} y^{14}
$$

Now, multiply these two results together:

$$
32 x^{15} y^{20} \times 9 x^{4} y^{14} = 288 x^{19} y^{34}
$$

### Step 2: Simplify the Denominator

The denominator is:

$$
6 x^{-5} y^{15}
$$

### Step 3: Combine Numerator and Denominator

Now, divide the simplified numerator by the denominator:

$$
\frac{288 x^{19} y^{34}}{6 x^{-5} y^{15}} 
$$

Simplify the constants and apply the laws of exponents:

$$
\frac{288}{6} = 48
$$

$$
x^{19} \div x^{-5} = x^{19 - (-5)} = x^{24}
$$

$$
y^{34} \div y^{15} = y^{19}
$$

### Final Simplified Expression

Putting it all together:

$$
48 x^{24} y^{19}
$$

**Answer:** After simplifying, the expression is 48 · x²⁴ · y¹⁹.
The simplified expression is $$ 48x^{24}y^{19} $$.

6. \( \frac{\left(2 x^{3} y^{4}\right)^{5}\left(3 x^{2} y^{7}\right)^{2}}{6 x^{-5} y^{15}} \)