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

Verified step by step guidance

1. Identify the expression to simplify: $(-2x^5{)}^5$.
2. Apply the power of a power property: $(a^m{)}^n=a^{m\cdot n}$.
3. Separate the expression into its components: $(-2{)}^5$ and $(x^5{)}^5$.
4. Simplify $(-2{)}^5$ by multiplying -2 by itself five times.
5. Simplify $(x^5{)}^5$ using the power of a power property: $x^{5\cdot 5}$.

Key Concepts

Exponent Rules

Exponent rules are fundamental principles that govern how to manipulate expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). Understanding these rules is essential for simplifying expressions with exponents.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/(a^n). This concept is crucial when simplifying expressions that may involve negative powers, ensuring that all terms are expressed in a standard form.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the given expression, (-2x^5)^5, recognizing it as a polynomial allows for the application of exponent rules to simplify it effectively.