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

Verified step by step guidance

1. Recognize that the expression is a power raised to another power: \((2^{2})^{5}\). According to the power of a power rule, when you raise a power to another power, you multiply the exponents.
2. Apply the power of a power rule: \((a^{m})^{n} = a^{m \times n}\). Here, \(a = 2\), \(m = 2\), and \(n = 5\).
3. Multiply the exponents: \(2 \times 5 = 10\).
4. Rewrite the expression using the product of exponents: \$2^{10}$.
5. The expression is now simplified to \$2^{10}$. You can leave it in this exponential form or calculate the numerical value if needed.

Key Concepts

Exponentiation Rules

Exponentiation rules govern how to simplify expressions involving powers. The power of a power rule states that (a^m)^n = a^(m*n), meaning you multiply the exponents when raising a power to another power.

Base and Exponent

In an expression like a^b, 'a' is the base and 'b' is the exponent. The exponent indicates how many times the base is multiplied by itself. Understanding this helps in correctly applying exponent rules.

Simplification of Numerical Expressions

Simplifying numerical expressions involves performing arithmetic operations and applying exponent rules to reduce the expression to its simplest form. For example, calculating powers of integers before applying further operations.