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

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Identify the rule for simplifying expressions of the form $(a^m)^n$, which states that you multiply the exponents: $(a^m)^n = a^{m \times n}$.

Apply this rule to the given expression $(x^{11})^5$ by multiplying the exponents 11 and 5.

Write the expression with the new exponent: $x^{11 \times 5}$.

Simplify the exponent multiplication to get $x^{55}$.

Express the final simplified form as $x^{55}$.

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주요 개념

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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.

Simplify each exponential expression in Exercises 23–64. (x11)5(x^{11})^5