Evaluate each exponential expression: (3^3)/(3^6)

Verified step by step guidance

1. Identify the base and the exponents in the expression: \( \frac{3^3}{3^6} \).
2. Apply the quotient rule for exponents, which states \( \frac{a^m}{a^n} = a^{m-n} \), where \( a \) is the base and \( m \) and \( n \) are the exponents.
3. Subtract the exponent in the denominator from the exponent in the numerator: \( 3^{3-6} \).
4. Simplify the expression by calculating the new exponent: \( 3^{-3} \).
5. Recognize that a negative exponent indicates a reciprocal, so \( 3^{-3} = \frac{1}{3^3} \).

Key Concepts

Exponential Expressions

Exponential expressions involve a base raised to a power, indicating how many times the base is multiplied by itself. For example, in the expression 3^3, the base is 3 and the exponent is 3, meaning 3 is multiplied by itself three times (3 × 3 × 3). Understanding how to manipulate these expressions is crucial for evaluating them correctly.

Laws of Exponents

The laws of exponents provide rules for simplifying expressions involving powers. One key rule is that when dividing two exponential expressions with the same base, you subtract the exponents: a^m / a^n = a^(m-n). This principle is essential for evaluating expressions like (3^3)/(3^6), as it allows for straightforward simplification.

Simplification of Fractions

Simplifying fractions involves reducing them to their simplest form, which can include canceling common factors. In the context of exponential expressions, this means applying the laws of exponents to rewrite the expression in a more manageable form. For instance, after applying the laws of exponents to (3^3)/(3^6), the result can be expressed as 1/3^3, illustrating the importance of simplification in mathematical evaluations.