Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (-2x⁵)⁵

Verified step by step guidance

1. Identify the expression to simplify: \((-2x^5)^5\).
2. Apply the power of a product rule, which states that \((ab)^n = a^n b^n\). So rewrite the expression as \((-2)^5 (x^5)^5\).
3. Use the power of a power rule for the variable part: \((x^5)^5 = x^{5 \times 5} = x^{25}\).
4. Calculate the power of the constant: \((-2)^5\) means multiplying -2 by itself 5 times, keeping track of the sign.
5. Combine the results to write the simplified expression as \((-2)^5 x^{25}\).

Key Concepts

Exponentiation of a Power

When raising a power to another power, multiply the exponents. For example, (x^a)^b = x^(a*b). This rule helps simplify expressions like (-2x^5)^5 by applying the exponent 5 to both the coefficient and the variable's exponent.

Properties of Exponents with Coefficients

When an expression with a coefficient and variable is raised to a power, raise both the coefficient and the variable to that power separately. For instance, (ab)^n = a^n * b^n. This allows simplification of (-2x^5)^5 into (-2)^5 * (x^5)^5.

Handling Negative Bases with Exponents

Raising a negative number to an odd power results in a negative number, while raising it to an even power results in a positive number. Since 5 is odd, (-2)^5 remains negative, which affects the sign of the simplified expression.