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 in exponents, 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 new exponent: \$2^{10}$.
5. The expression is now simplified to \$2^{10}$. You can leave it in this exponential form unless you are asked to calculate the numerical value.

Key Concepts

Exponentiation and Power Rules

Exponentiation involves raising a base to a power, and power rules help simplify expressions with exponents. The key rule here is (a^m)^n = a^(m*n), which means when raising a power to another power, multiply the exponents.

Properties of Real Numbers

Understanding that variables represent nonzero real numbers ensures that operations like exponentiation are valid and that no division by zero or undefined expressions occur. This context allows safe application of exponent rules.

Simplification of Expressions

Simplification involves rewriting expressions in a more compact or standard form without changing their value. Applying exponent rules correctly reduces complex expressions like (2²)⁵ to a single power, making calculations easier.