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The question asks you to select all expressions that are equivalent to \( 3 \times 3^5 \).

We can start by simplifying \( 3 \times 3^5 \) using the property of exponents, \( a^m \times a^n = a^{m+n} \). Here, we have:

\[
3 \times 3^5 = 3^1 \times 3^5 = 3^{1+5} = 3^6
\]

Now, let's analyze the options to see which ones are equivalent to \( 3^6 \):

1. \( \frac{1}{3^6} \): This is equivalent to \( 3^{-6} \), which is not equal to \( 3^6 \).
2. \( 12^6 \div 4^6 \): This can be simplified as \( \left(\frac{12}{4}\right)^6 = 3^6 \), which is indeed equal to \( 3^6 \).
3. \( (3^1)^4 \): Simplifies to \( 3^{1 \times 4} = 3^4 \), which is not equal to \( 3^6 \).
4. \( \frac{1}{3^3} \): This is equivalent to \( 3^{-3} \), which is not equal to \( 3^6 \).

From this analysis, the correct expressions equivalent to \( 3^6 \) are:

- \( 12^6 \div 4^6 \)

It seems you've selected all options except for \( (3^1)^4 \), which is correct because that option does not equal \( 3^6 \).

Do you have any questions or would you like more details? Here are some related questions you might find interesting:

1. How do you simplify expressions using the power rule?
2. What are the common mistakes when working with exponents?
3. How can you recognize equivalent expressions in different forms?
4. What is the difference between positive and negative exponents?
5. How can you apply exponent rules to solve real-world problems?

**Tip:** Always double-check your work by simplifying expressions step-by-step to ensure accuracy.

Select all the expressions that are equivalent to 3 × 3^5.

Solve the problem by identifying all expressions equivalent to 3 × 3^5 using exponent rules. This involves simplifying exponents and understanding key properties of exponentiation. Suitable for 7th grade students.

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