Question

negative / zero exponents (level 1)
score: 0/4 penalty: none
question
which expression is equivalent to $3^{-5} \times 3^{4}$?
answer
\\( 3 \\) \\( 1 \\) \\( \frac{1}{3} \\) \\( 3^{9} \\)
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Explanation:

Step1: Recall exponent rule

When multiplying exponents with the same base, we use the rule \(a^m \times a^n = a^{m + n}\). Here, the base \(a = 3\), \(m=-5\), and \(n = 4\) (wait, wait, the second exponent is 4? Wait, the problem is \(3^{-5} \times 3^{4}\)? Wait, let me check again. Wait, the user's problem: "Which expression is equivalent to \(3^{-5} \times 3^{4}\)?" Wait, maybe a typo, but assuming it's \(3^{-5} \times 3^{4}\), then apply the rule.

Wait, no, the original problem: "Which expression is equivalent to \(3^{-5} \times 3^{4}\)?" Wait, let's compute the exponent: \(-5 + 4=-1\)? No, wait, maybe the second exponent is 4? Wait, no, maybe I misread. Wait, the options are 3, 1, \(1/3\), \(3^9\). Wait, maybe the second exponent is 5? No, wait, let's re-express. Wait, \(3^{-5} \times 3^{4}\). Using \(a^m \times a^n = a^{m + n}\), so \(3^{-5 + 4}=3^{-1}\). And \(3^{-1}=\frac{1}{3}\). Wait, but let's check the options. The options are 3, 1, \(1/3\), \(3^9\). So \(3^{-5} \times 3^{4}=3^{-5 + 4}=3^{-1}=\frac{1}{3}\).

Wait, maybe the problem was \(3^{-5} \times 3^{5}\)? Then it would be \(3^{0}=1\), but the options have \(1/3\) and 1. Wait, let's check the original problem again. The user's image: "Which expression is equivalent to \(3^{-5} \times 3^{4}\)?" Wait, maybe a typo, but let's proceed with the given.

Step2: Apply the exponent rule

For \(3^{-5} \times 3^{4}\), using \(a^m \times a^n = a^{m + n}\), so \(3^{-5 + 4}=3^{-1}\).

Step3: Simplify \(3^{-1}\)

Recall that \(a^{-n}=\frac{1}{a^n}\), so \(3^{-1}=\frac{1}{3}\).

Answer:

\(\frac{1}{3}\) (the option with \(1/3\))