Questions: (75xy)(5x^2y^-5)^2

Solution Steps

To simplify the given expression, we need to follow these steps:

1. Distribute the exponent in the second term to both \(x\) and \(y\).
2. Multiply the coefficients and combine the exponents of like bases using the laws of exponents.

We start with the expression \( (75xy)\left(5x^{2}y^{-5}\right)^{2} \). First, we distribute the exponent in the second term:

\[ \left(5x^{2}y^{-5}\right)^{2} = 5^{2} \cdot (x^{2})^{2} \cdot (y^{-5})^{2} = 25 \cdot x^{4} \cdot y^{-10} \]

Next, we multiply the first term \( 75xy \) by the result from Step 1:

\[ (75xy) \cdot (25x^{4}y^{-10}) = 75 \cdot 25 \cdot x^{1} \cdot x^{4} \cdot y^{1} \cdot y^{-10} \]

Now, we combine the coefficients and the exponents of like bases:

• Coefficient: \( 75 \cdot 25 = 1875 \)
• For \( x \): \( x^{1} \cdot x^{4} = x^{1+4} = x^{5} \)
• For \( y \): \( y^{1} \cdot y^{-10} = y^{1-10} = y^{-9} \)

Thus, the expression simplifies to:

\[ 1875 \cdot x^{5} \cdot y^{-9} \]

Final Answer