Question

SIMPLIFYING EXPRESSIONS Simplify. Write your answer as a power or as an expression containing powers. $$\left(-3 a^{2} b^{2}\right)^{3}$$

Short answer

The simplified expression for \(\left(-3 a^{2} b^{2}\right)^{3}\) is \(-27a^6b^6\).

Step-by-step solution

Recognize and observe order of operation and apply it to the expression.

Observe that the expression \(\left(-3 a^{2} b^{2}\right)^{3}\) is a product which is raised to power 3. The order of operation dictates we first simplify what's inside the parenthesis, but in this case it cannot be further simplified. So, we proceed to raising the expression to power 3.

Apply the rule of power raised to a power.

When raising a product to an exponent, each element inside the parentheses is raised to that exponent. Therefore, \((-3)^3\), \((a^2)^3\), and \((b^2)^3\) need to be calculated separately.

Simplify to obtain final expression.

Calculate \( (-3)^3 \), \((a^2)^3\), and \((b^2)^3\) separately. For \( (-3)^3 \), the result is -27, as the cube of a negative number is negative. For \((a^2)^3\), the result is \(a^{2*3}\) which simplifies to \(a^6\), as the exponent multiplies when a power is raised to a power. Similarly, for \((b^2)^3\), the result is \(b^{2*3}\) which simplifies to \(b^6\).

Combine results.

Combining the results \(-27\), \(a^6\) and \(b^6\), we get the simplified expression as \(-27a^6b^6\). Write this as the final answer.