Question

simplify the expression.
$(-3x^2y^2)^3$

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). For the expression \((-3x^{2}y^{2})^{3}\), we can apply this rule to each factor inside the parentheses. So we have \((-3)^3 \cdot (x^{2})^3 \cdot (y^{2})^3\).

Step2: Simplify each factor

• For \((-3)^3\): We know that \((-3)\times(-3)\times(-3)= -27\).
• For \((x^{2})^3\): Using the power - of - a - power rule \((a^m)^n=a^{m\times n}\), here \(a = x\), \(m = 2\) and \(n = 3\), so \((x^{2})^3=x^{2\times3}=x^{6}\).
• For \((y^{2})^3\): Similarly, using the power - of - a - power rule, with \(a = y\), \(m = 2\) and \(n = 3\), we get \((y^{2})^3=y^{2\times3}=y^{6}\).

Step3: Multiply the simplified factors together

Now we multiply the three simplified factors: \((-3)^3\cdot(x^{2})^3\cdot(y^{2})^3=-27\times x^{6}\times y^{6}=-27x^{6}y^{6}\).

Answer:

\(-27x^{6}y^{6}\)