Simplify each exponential expression in Exercises 23–64. $(x^{11}{)}^5$

Verified step by step guidance

1. Identify the rule for simplifying expressions of the form \((a^m)^n\), which states that you multiply the exponents: \((a^m)^n = a^{m \times n}\).
2. Apply this rule to the given expression \((x^{11})^5\) by multiplying the exponents 11 and 5.
3. Write the expression with the new exponent: \(x^{11 \times 5}\).
4. Simplify the exponent multiplication to get \(x^{55}\).
5. Express the final simplified form as \(x^{55}\).

Key Concepts

Laws of Exponents

The laws of exponents provide rules for simplifying expressions involving powers. One key rule is that when raising a power to another power, you multiply the exponents. For example, (x^a)^b = x^(a*b). Understanding these laws is essential for simplifying expressions like (x^11)^5.

Power of a Power Rule

The power of a power rule states that when an exponential expression is raised to another exponent, you multiply the exponents. This rule helps simplify expressions by reducing nested exponents into a single exponent, making calculations and further manipulations easier.

Variable Exponents

Variable exponents involve expressions where the base is a variable raised to a power. Recognizing how to manipulate these exponents correctly, especially when combined with exponent rules, is crucial for simplifying algebraic expressions and solving related problems.