Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. 6⁴/6^-2

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{6^{4}}{6^{-2}}\).
2. Recall the quotient rule for exponents: \(\frac{a^{m}}{a^{n}} = a^{m-n}\), where \(a\) is a nonzero base.
3. Apply the quotient rule to the expression: \$6^{4 - (-2)}$.
4. Simplify the exponent by subtracting the exponents: \$4 - (-2) = 4 + 2$.
5. Rewrite the expression with the simplified exponent: \$6^{6}$. This expression has no negative exponents.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify 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 6^4 / 6^-2.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent, such as a^(-n) = 1 / a^n. To write answers without negative exponents, convert any negative powers into their reciprocal form, ensuring the expression contains only positive exponents.

Simplification of Expressions

Simplification involves rewriting expressions in their simplest form by applying algebraic rules. This includes combining like terms, reducing powers using exponent laws, and eliminating negative exponents. Simplifying makes expressions easier to understand and use in further calculations.