Question

select all the expressions that are equivalent to $8^{6} \cdot 8^{-3}$. $8^{-18}$ $\frac{1}{8^{-3}}$ $\frac{1}{8^{3}}$ $\frac{1}{8^{-18}}$

Explanation:

Step1: Simplify \(8^{6}\cdot8^{-3}\) using exponent rule

When multiplying exponents with the same base, we add the exponents: \(a^{m}\cdot a^{n}=a^{m + n}\). So \(8^{6}\cdot8^{-3}=8^{6+( - 3)}=8^{3}\).

Step2: Analyze each option

- Option \(8^{-18}\): This is \(8^{-18} eq8^{3}\), so it is not equivalent.
- Option \(\frac{1}{8^{-3}}\): Using the rule \(a^{-n}=\frac{1}{a^{n}}\), so \(\frac{1}{8^{-3}} = 8^{3}\), which is equivalent.
- Option \(\frac{1}{8^{3}}\): This is \(8^{-3} eq8^{3}\), wait, no, wait: Wait, \(8^{3}=\frac{1}{8^{-3}}\) and also, \(8^{3}=\frac{1}{\frac{1}{8^{3}}}\)? Wait, no, let's re - evaluate. Wait, \(8^{6}\cdot8^{-3}=8^{3}\). Now, \(\frac{1}{8^{-3}}=8^{3}\) (because \(a^{-n}=\frac{1}{a^{n}}\), so \(\frac{1}{a^{-n}}=a^{n}\)). And \(\frac{1}{8^{3}}=8^{-3}\), which is not equal to \(8^{3}\). Wait, there was a mistake in the initial check. Let's correct:

Wait, \(8^{6}\cdot8^{-3}=8^{6 - 3}=8^{3}\).

Now, let's analyze each option:

1. \(8^{-18}\): \(8^{-18}\) is not equal to \(8^{3}\), so it's incorrect.
2. \(\frac{1}{8^{-3}}\): By the negative exponent rule \(a^{-n}=\frac{1}{a^{n}}\), so \(\frac{1}{8^{-3}}=8^{3}\) (since \(\frac{1}{\frac{1}{8^{3}}}=8^{3}\)), so this is equivalent.
3. \(\frac{1}{8^{3}}\): \(\frac{1}{8^{3}} = 8^{-3}\), which is not equal to \(8^{3}\), so it's incorrect. Wait, no, wait, maybe I made a mistake. Wait, \(8^{3}=\frac{1}{8^{-3}}\), and also, \(8^{3}\) can be written as \(\frac{1}{8^{-3}}\), but what about \(\frac{1}{8^{3}}\)? No, \(\frac{1}{8^{3}}\) is \(8^{-3}\). Wait, but the original problem's initial check was wrong. Let's start over.

Correct Step 2:

We have \(8^{6}\cdot8^{-3}=8^{6 + (-3)}=8^{3}\).

Now check each option:

- \(8^{-18}\): Not equal to \(8^{3}\).
- \(\frac{1}{8^{-3}}\): Using \(a^{-n}=\frac{1}{a^{n}}\), so \(\frac{1}{8^{-3}} = 8^{3}\) (because \(\frac{1}{8^{-3}}=8^{3}\) as per \(a^{-n}=\frac{1}{a^{n}}\) implies \(\frac{1}{a^{-n}}=a^{n}\)). So this is equivalent.
- \(\frac{1}{8^{3}}\): This is \(8^{-3}\), not equal to \(8^{3}\), so it's incorrect.
- \(\frac{1}{8^{-18}}\): This is \(8^{18}\), not equal to \(8^{3}\), so it's incorrect.

Wait, the initial selection in the problem was wrong. The correct equivalent expressions are \(\frac{1}{8^{-3}}\) and also, let's see, \(8^{3}\) is equal to itself, but among the given options, only \(\frac{1}{8^{-3}}\) is equivalent to \(8^{6}\cdot8^{-3}\) (since \(8^{6}\cdot8^{-3}=8^{3}=\frac{1}{8^{-3}}\)). Wait, maybe there was a miscalculation earlier. Let's re - do:

\(8^{6}\times8^{-3}=8^{6 - 3}=8^{3}\)

Now:

- \(8^{-18}\): No.
- \(\frac{1}{8^{-3}}\): \(8^{-3}=\frac{1}{8^{3}}\), so \(\frac{1}{8^{-3}} = 8^{3}\), yes.
- \(\frac{1}{8^{3}}\): \(8^{-3}\), no.
- \(\frac{1}{8^{-18}}\): \(8^{18}\), no.

Wait, the original problem's check marks were wrong. So the correct equivalent expression is \(\frac{1}{8^{-3}}\). But maybe I misread the options. Let's check again.

Wait, the options are:

1. \(8^{-18}\)
2. \(\frac{1}{8^{-3}}\)
3. \(\frac{1}{8^{3}}\)
4. \(\frac{1}{8^{-18}}\)

We have \(8^{6}\cdot8^{-3}=8^{3}\).

\(\frac{1}{8^{-3}}=8^{3}\) (because \(a^{-n}=\frac{1}{a^{n}}\) so \(\frac{1}{a^{-n}}=a^{n}\)).

So the only equivalent expression among the options is \(\frac{1}{8^{-3}}\). But maybe there was a mistake in the problem's initial check.

Wait, maybe the user made a mistake in the initial selection. Let's solve it properly:

Given \(8^{6}\cdot8^{-3}\)

Using the product of exponents rule \(a^{m}\cdot a^{n}=a^{m + n}\), we get \(8^{6+( - 3)}=8^{3}\)

Now, let's analyze each option:

- \(8^{-18}\): \(8^{-18}\) is not equal to \(8^{3}\), so it's incorrect.
- \(\frac{1}{8^{-3}}\): By the rule \(a^{-n}=\frac{1}{a^{n}}\), so \(\frac{1}{8^{-3}} = 8^{3}\) (since \(\frac{1}{\frac{1}{8^{3}}}=8^{3}\)), so this is correct.
- \(\frac{1}{8^{3}}\): This is \(8^{-3}\), which is not equal to \(8^{3}\), so it's incorrect.
- \(\frac{1}{8^{-18}}\): This is \(8^{18}\), which is not equal to \(8^{3}\), so it's incorrect.

So the correct answer should be \(\frac{1}{8^{-3}}\) (the second option).

Answer:

\(\frac{1}{8^{-3}}\) (the option \(\boldsymbol{\frac{1}{8^{-3}}}\))