Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3.(2²)⁵

Verified step by step guidance

1. Identify the base and the exponents in the expression \((2^2)^5\).
2. Apply the power of a power property of exponents, which states \((a^m)^n = a^{m \cdot n}\).
3. Substitute the values into the property: \((2^2)^5 = 2^{2 \cdot 5}\).
4. Calculate the new exponent by multiplying: \(2 \cdot 5\).
5. Rewrite the expression with the simplified exponent: \(2^{10}\).

Key Concepts

Exponents and Powers

Exponents represent repeated multiplication of a base number. The expression a^n means 'a multiplied by itself n times.' Understanding the laws of exponents, such as the power of a power rule (a^(m*n) = (a^m)^n), is crucial for simplifying expressions involving exponents.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed to ensure consistent results. The common acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps remember this order, emphasizing that exponents should be calculated before multiplication or addition.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form while maintaining its value. This often includes combining like terms, applying exponent rules, and eliminating unnecessary parentheses. In the case of (2²)⁵, applying the power of a power rule simplifies the expression effectively.