We are given the expression $13^6 \cdot 13^6 \cdot (13^0)$, and we need to find which options from A to F are equivalent to this expression. Let's break this down step-by-step:

Simplifying the expression:

We can combine the exponents of the powers of 13 by using the property of exponents:

$a^m \cdot a^n = a^{m+n}$

In this case, $13^6 \cdot 13^6 = 13^{6+6} = 13^{12}$.

Next, multiply by $13^0$. We know from the zero exponent rule that:

$a^0 = 1$

Therefore, multiplying by $13^0$ is equivalent to multiplying by 1, so it doesn't affect the result. Thus:

$13^{12} \cdot 13^0 = 13^{12}$

The expression simplifies to $13^{12}$.

Analyzing the options:

• A. $13^{5+6}$: This simplifies to $13^{11}$, which is not correct.

• B. $13^{3 \cdot 2 - 6}$: Simplifying the exponent:

  $13^{3 \cdot 2 - 6} = 13^{6 - 6} = 13^0$

  This is equal to $1$, which is not correct.

• C. $(13^{11})^0$: Any expression to the power of 0 is 1, so this simplifies to $1$, which is not correct.

• D. $(13^{12})^0$: This also simplifies to $1$, which is not correct.

• E. $13^{11} \cdot 13^0$: Using the property of exponents:

  $13^{11} \cdot 13^0 = 13^{11}$

  This is not correct, as we need $13^{12}$.

• F. $13^{12} \cdot 13^0$: This simplifies to $13^{12}$, which is correct.

Correct answer:

• Only F is equivalent to the given expression.

Would you like more details on any step or have any questions?

Here are 5 related questions for further exploration:

1. What is the result of $a^m \cdot a^n$ when $m$ and $n$ are negative?
2. How do you simplify expressions with exponents when one of the terms is raised to 0?
3. Can you explain why $a^0 = 1$ for any non-zero $a$?
4. How do you simplify expressions where terms with the same base are divided?
5. What happens when an exponentiation is raised to another power, like $(a^m)^n$?

Tip: Always remember to simplify expressions with zero exponents carefully, as they often reduce to 1, affecting the rest of the equation.