Simplify each expression. Assume all variables represent nonzero real numbers. (6⁴)³

Verified step by step guidance

1. Identify the expression given: \( (6^{4})^{3} \). This is a power raised to another power.
2. Recall the exponent rule for powers raised to powers: \( (a^{m})^{n} = a^{m \times n} \).
3. Apply this rule to the expression: multiply the exponents 4 and 3 to get a single exponent.
4. Rewrite the expression as \( 6^{4 \times 3} \).
5. Simplify the exponent multiplication to get \( 6^{12} \), which is the simplified form of the expression.

Key Concepts

Laws of Exponents

The laws of exponents provide rules for simplifying expressions involving powers. For example, when raising a power to another power, multiply the exponents. This helps simplify expressions like (6^4)^3 by calculating 6^(4×3) = 6^12.

Power of a Power Rule

The power of a power rule states that (a^m)^n = a^(m×n), where a is a nonzero base and m, n are exponents. This rule is essential for simplifying nested exponents, ensuring the expression is rewritten with a single exponent.

Properties of Nonzero Real Numbers

Assuming variables or bases are nonzero real numbers ensures that exponent rules apply without exceptions like division by zero. This assumption allows safe use of exponent laws and avoids undefined expressions.