Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. (1/3)^-2

Verified step by step guidance

1. Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\), where \(a\) is a nonzero number and \(n\) is a positive integer.
2. Apply this rule to the expression \(\left(\frac{1}{3}\right)^{-2}\). This means you take the reciprocal of \(\frac{1}{3}\) and raise it to the positive exponent 2.
3. Rewrite \(\left(\frac{1}{3}\right)^{-2}\) as \(\left(\frac{3}{1}\right)^2\) by flipping the fraction inside the parentheses.
4. Now, simplify \(\left(\frac{3}{1}\right)^2\) by squaring both the numerator and the denominator: \(\frac{3^2}{1^2}\).
5. Calculate the powers: \(3^2 = 9\) and \(1^2 = 1\), so the expression simplifies to \(\frac{9}{1}\), which is just 9.

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a^(-n) equals 1 divided by a^n, where a is a nonzero number. This rule allows rewriting expressions without negative exponents.

Evaluating Powers of Fractions

When raising a fraction to a power, both the numerator and denominator are raised to that power separately. For instance, (1/3)^2 equals 1^2 divided by 3^2, which simplifies to 1/9. This helps in simplifying fractional expressions with exponents.

Assumption of Nonzero Variables

Assuming variables are nonzero ensures that expressions involving division or negative exponents are valid, as division by zero is undefined. This assumption is crucial when rewriting expressions with negative exponents to avoid undefined operations.