Questions: Simplify. (-3x^3y^3z^3)^4(-2y^4z^2)^2

Solution Steps

First, apply the power of a product rule, which states that \((ab)^n = a^n b^n\), to each term in the expression:

\[ \left(-3 x^{3} y^{3} z^{3}\right)^{4} = (-3)^4 (x^3)^4 (y^3)^4 (z^3)^4 \]

\[ \left(-2 y^{4} z^{2}\right)^{2} = (-2)^2 (y^4)^2 (z^2)^2 \]

Calculate each power:

\[ (-3)^4 = 81, \quad (x^3)^4 = x^{12}, \quad (y^3)^4 = y^{12}, \quad (z^3)^4 = z^{12} \]

\[ (-2)^2 = 4, \quad (y^4)^2 = y^8, \quad (z^2)^2 = z^4 \]

Combine the results from Step 2:

\[ 81 x^{12} y^{12} z^{12} \cdot 4 y^8 z^4 \]

Multiply the coefficients and combine the like terms by adding the exponents of the same base:

\[ 81 \times 4 = 324 \]

\[ x^{12}, \quad y^{12} \cdot y^8 = y^{20}, \quad z^{12} \cdot z^4 = z^{16} \]

Final Answer