Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (6^4)^3

Verified step by step guidance

1. Identify the expression to simplify: $(6^4{)}^3$.
2. Apply the power of a power property of exponents, which states $(a^m{)}^n=a^{m\cdot n}$.
3. In this case, set $a=6$, $m=4$, and $n=3$.
4. Multiply the exponents: $4\cdot 3$.
5. Rewrite the expression as $6^{4\cdot 3}$.

Key Concepts

Exponent Rules

Exponent rules are mathematical guidelines that dictate how to handle expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)), the power of a power ((a^m)^n = a^(m*n)), and the power of a product ((ab)^n = a^n * b^n). Understanding these rules is essential for simplifying expressions with exponents.

Order of Operations

The order of operations is a set of rules that determines the sequence in which calculations are performed in a mathematical expression. The common acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps remember this order. Following these rules ensures that expressions are simplified correctly and consistently.

Nonzero Real Numbers

Nonzero real numbers are all the real numbers except for zero, including both positive and negative values. This distinction is important in algebra because certain operations, like division, are undefined when zero is involved. When simplifying expressions, assuming variables represent nonzero real numbers allows for the application of exponent rules without concern for undefined behavior.