Question

multiplying with like bases
choose three correct answers that are equal to $6^{4} (6^{-5})$.
options: $\frac{1}{6^{-1}}$, $\frac{1}{6}$, $\frac{1}{6^{1}}$, $\frac{1}{-6}$, $6^{-1}$

Explanation:

Step1: Simplify the expression

We start with the expression \(6^{4}\times6^{-5}\). When multiplying exponents with the same base, we use the rule \(a^{m}\times a^{n}=a^{m + n}\). So, \(6^{4}\times6^{-5}=6^{4+( - 5)}=6^{-1}\).

Step2: Recall the definition of negative exponents

The rule for negative exponents is \(a^{-n}=\frac{1}{a^{n}}\). So, \(6^{-1}=\frac{1}{6^{1}}=\frac{1}{6}\). Also, \(6^{-1}\) can be written as \(\frac{1}{6^{-(-1)}}\)? Wait, no. Wait, \(6^{-1}=\frac{1}{6^{1}}\), and also \(\frac{1}{6^{-(-1)}}\) is not correct. Wait, let's re - express:

We know that \(6^{-1}=\frac{1}{6}\), and also \(6^{-1}=\frac{1}{6^{1}}\), and \(\frac{1}{6^{-(-1)}}\) is wrong. Wait, let's check each option:

• Option 1: \(\frac{1}{6}\). Since \(6^{-1}=\frac{1}{6}\), this is correct.
• Option 2: \(\frac{1}{6^{-1}}\). \(\frac{1}{6^{-1}}=6^{1} = 6\), which is not equal to \(6^{-1}\), so this is incorrect. Wait, no, wait our original expression was \(6^{4}\times6^{-5}=6^{-1}\). Let's re - evaluate:

Wait, \(6^{4}\times6^{-5}=6^{4 - 5}=6^{-1}\). Now let's check each option:

1. \(\frac{1}{6}\): Since \(6^{-1}=\frac{1}{6}\), this is equal.
2. \(\frac{1}{6^{-1}}\): \(\frac{1}{6^{-1}}=6^{1}=6

eq6^{-1}\), so no.

1. \(\frac{1}{-6}\): This is \(-\frac{1}{6}

eq6^{-1}\), so no.

1. \(\frac{1}{6^{1}}\): Since \(6^{1} = 6\), \(\frac{1}{6^{1}}=\frac{1}{6}=6^{-1}\), so this is correct.
2. \(6^{-1}\): This is the simplified form of our original expression, so this is correct.

Answer:

The correct answers are \(\frac{1}{6}\), \(\frac{1}{6^{1}}\), \(6^{-1}\) (corresponding to the first, fourth, and fifth options in the given layout).