The problem asks to identify expressions that are equivalent to $7^7 \times 7^7$.

To solve this:

1. Simplify the given expression: $7^7 \times 7^7$.
   • Using the property of exponents: $a^m \times a^n = a^{m+n}$,
   • So, $7^7 \times 7^7 = 7^{7+7} = 7^{14}$.

Now, let's evaluate each of the options:

• Option 1: $\frac{1}{7^{14}}$ is equivalent to $7^{-14}$, which is not equal to $7^{14}$.
• Option 2: $7^{49}$ is not equal to $7^{14}$.
• Option 3: $\left(7^7\right)^2$ can be simplified using the power of a power rule: $(a^m)^n = a^{m \times n}$, so $\left(7^7\right)^2 = 7^{7 \times 2} = 7^{14}$.
• Option 4: $7^{14}$ is exactly $7^{14}$.

Thus, the correct expressions that are equivalent to $7^7 \times 7^7$ are:

• $\left(7^7\right)^2$
• $7^{14}$

Would you like more details on exponent rules or any related questions? Here are five questions to explore further:

1. What are the different properties of exponents?
2. How do you simplify expressions involving exponents with different bases?
3. How would the answer change if the problem was $7^7 \times 7^6$?
4. How do you solve equations that involve exponents?
5. What is the difference between $7^{-14}$ and $7^{14}$?

Tip: When multiplying terms with the same base, simply add the exponents to find the product.