Simplify each expression. Assume all variables represent nonzero real numbers. (-2x⁵)⁵

Verified step by step guidance

1. Identify the expression to simplify: \((-2x^{5})^{5}\). This means the entire quantity \(-2x^{5}\) is raised to the 5th power.
2. Apply the power of a product rule, which states that \((ab)^{n} = a^{n} b^{n}\). So rewrite the expression as \((-2)^{5} \cdot (x^{5})^{5}\).
3. Calculate the power of the coefficient: \((-2)^{5}\). Remember that raising a negative number to an odd power keeps it negative.
4. Use the power of a power rule for the variable part: \((x^{5})^{5} = x^{5 \cdot 5} = x^{25}\).
5. Combine the results to write the simplified expression as \((-2)^{5} \cdot x^{25}\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. For example, when raising a power to another power, multiply the exponents (i.e., (a^m)^n = a^(m*n)). This rule is essential for simplifying expressions like (-2x^5)^5.

Distributive Property of Exponents over Multiplication

When an expression inside parentheses is raised to a power, the exponent applies to each factor inside. For instance, (ab)^n = a^n * b^n. This helps in breaking down (-2x^5)^5 into (-2)^5 * (x^5)^5 for easier simplification.

Handling Negative Bases and Variables

When raising a negative number to a power, the sign depends on whether the exponent is even or odd. An odd exponent keeps the negative sign, while an even exponent makes it positive. Also, variables raised to powers follow exponent rules, assuming variables are nonzero.