Simplify each exponential expression: $(-2x^4{y}^3{)}^3$

Verified step by step guidance

1. Identify the expression to simplify: \((-2x^{4}y^{3})^{3}\). This means the entire quantity inside the parentheses is raised to the power of 3.
2. Apply the power of a product rule, which states that \((abc)^{n} = a^{n}b^{n}c^{n}\). So, rewrite the expression as \((-2)^{3} \cdot (x^{4})^{3} \cdot (y^{3})^{3}\).
3. Use the power of a power rule for each variable term: \((x^{m})^{n} = x^{m \cdot n}\). So, \((x^{4})^{3} = x^{4 \cdot 3}\) and \((y^{3})^{3} = y^{3 \cdot 3}\).
4. Calculate the exponents for the variables: \(x^{12}\) and \(y^{9}\), but do not multiply the coefficients yet.
5. Combine all parts to write the simplified expression as \((-2)^{3} \cdot x^{12} \cdot y^{9}\). This is the fully simplified form before calculating the numerical coefficient.

Key Concepts

Exponentiation of a Product

When raising a product to a power, each factor inside the parentheses is raised to that power separately. For example, (abc)^n = a^n * b^n * c^n. This rule helps simplify expressions like (-2x^4y^3)^3 by applying the exponent to -2, x^4, and y^3 individually.

Power of a Power Rule

The power of a power rule states that (a^m)^n = a^(m*n). This means when an exponent is raised to another exponent, you multiply the exponents. In the expression (-2x^4y^3)^3, this rule is used to simplify x^(4*3) and y^(3*3).

Handling Negative Bases with Exponents

When a negative number is raised to an odd power, the result remains negative; if raised to an even power, the result is positive. For (-2)^3, since 3 is odd, the result is -8. This concept is important to correctly simplify the coefficient in the expression.