Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 4⁸/4⁶

Verified step by step guidance

1. Recall the quotient rule for exponents, which states that for any nonzero base \(a\) and integers \(m\) and \(n\), \(\frac{a^m}{a^n} = a^{m-n}\).
2. Apply the quotient rule to the given expression \(\frac{4^8}{4^6}\) by subtracting the exponents: \$4^{8-6}$.
3. Simplify the exponent subtraction: \$8 - 6 = 2\(, so the expression becomes \)4^2$.
4. Rewrite the expression without negative exponents (already done here since the exponent is positive).
5. Express the final simplified form as \$4^2\(, which can also be written as \)16$ if numerical evaluation is desired.

Key Concepts

Laws of Exponents

The laws of exponents provide rules for simplifying expressions involving powers. For division, subtract the exponent of the denominator from the exponent of the numerator when the bases are the same, i.e., 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 requires answers without negative exponents, understanding how to rewrite expressions to avoid negative powers is crucial.

Properties of Nonzero Real Numbers

Assuming variables represent nonzero real numbers ensures that division by zero does not occur and that exponent rules apply correctly. This assumption allows simplification without restrictions and avoids undefined expressions.