Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (6⁴)³

Verified step by step guidance

1. Recall the power of a power property of exponents, which states that \((a^m)^n = a^{m \cdot n}\), where \(a\) is a base and \(m\), \(n\) are exponents.
2. Identify the base and the exponents in the expression \((6^4)^3\). Here, the base is 6, the inner exponent is 4, and the outer exponent is 3.
3. Apply the power of a power rule by multiplying the exponents: \$4 \times 3$.
4. Rewrite the expression as a single power: \$6^{4 \times 3}$.
5. Simplify the exponent multiplication to get \$6^{12}$, which is the simplified form of the original expression.

Key Concepts

Exponentiation Rules

Exponentiation rules govern how to simplify expressions involving powers. One key rule is that when raising a power to another power, you multiply the exponents. For example, (a^m)^n = a^(m*n). This rule helps simplify expressions like (6^4)^3.

Properties of Real Numbers

Understanding properties of real numbers, such as nonzero values, ensures that operations like exponentiation are valid and avoid undefined expressions. Since variables represent nonzero real numbers, division and exponentiation are well-defined, allowing simplification without restrictions.

Simplification of Expressions

Simplification involves rewriting expressions in a simpler or more compact form without changing their value. Applying exponent rules correctly reduces complex expressions like (6^4)^3 to a single power, making calculations easier and expressions clearer.