https://file.aiquickdraw.com/mathbot/file/1726249968503n2ehwbwh.jpg

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.

Which expressions are equivalent to 13^6 ⋅ 13^6 ⋅ (13^0)? Select all that apply.

Learn how to simplify exponential expressions like 13^6 ⋅ 13^6 ⋅ (13^0). This problem focuses on exponent rules such as adding exponents with the same base and applying the zero exponent theorem. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation