Questions: Simplify. [ left(-2 x^3 y^4 zright)left(-x^3 y^2 zright)^4 ]

Solution Steps

To simplify the given expression, we need to apply the laws of exponents. First, simplify the expression inside the parentheses by raising each term to the power of 4. Then, multiply the resulting expression by the first term. Combine like terms by adding the exponents of the same base.

First, simplify the expression \((-x^3 y^2 z)^4\). Apply the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\). This gives: \[ (-x^3 y^2 z)^4 = (-1)^4 \cdot x^{3 \cdot 4} \cdot y^{2 \cdot 4} \cdot z^4 = x^{12} y^8 z^4 \]

Now, multiply the simplified expression from Step 1 by the first term \(-2x^3y^4z\): \[ (-2x^3y^4z) \cdot (x^{12}y^8z^4) \]

Combine the terms by adding the exponents of the same base:

• For \(x\): \(x^{3+12} = x^{15}\)
• For \(y\): \(y^{4+8} = y^{12}\)
• For \(z\): \(z^{1+4} = z^5\)

Thus, the expression becomes: \[ -2 \cdot x^{15} \cdot y^{12} \cdot z^5 \]

Final Answer