Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. 4⁸/4⁶

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{4^{8}}{4^{6}}\).
2. Recall the quotient rule for exponents, which states that for the same base \(a\), \(\frac{a^{m}}{a^{n}} = a^{m-n}\).
3. Apply the quotient rule to the expression: \(\frac{4^{8}}{4^{6}} = 4^{8-6}\).
4. Simplify the exponent by subtracting: \$4^{8-6} = 4^{2}$.
5. Write the final expression without negative exponents (already done here): \$4^{2}$.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. For division, subtract the exponent in the denominator from the exponent in the numerator when the bases are the same, e.g., a^m / a^n = a^(m-n). This rule is essential for simplifying expressions like 4^8 / 4^6.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent, such as a^(-n) = 1 / a^n. Since the problem asks for answers without negative exponents, it is important to rewrite any negative exponents as positive by using this reciprocal rule.

Properties of Real Numbers

Understanding that variables represent nonzero real numbers ensures that division by zero does not occur and that exponent rules apply correctly. This assumption allows simplification without concern for undefined expressions or zero bases.