https://file.aiquickdraw.com/mathbot/file/1730030550645uhyijidz.jpg

The problem given is:

\[
\frac{2^m \times 2^6 \times 2^{-4}}{2^{-5}} = 2^{12}
\]

To solve for \( m \), we can use the properties of exponents:

1. **Combine the exponents in the numerator**:
   \[
   2^m \times 2^6 \times 2^{-4} = 2^{m + 6 - 4} = 2^{m + 2}
   \]

2. **Simplify the division by using exponent rules**:
   \[
   \frac{2^{m + 2}}{2^{-5}} = 2^{(m + 2) - (-5)} = 2^{m + 2 + 5} = 2^{m + 7}
   \]

3. **Set up the equation**:
   Since this expression equals \( 2^{12} \), we have:
   \[
   2^{m + 7} = 2^{12}
   \]

4. **Equate the exponents**:
   \[
   m + 7 = 12
   \]

5. **Solve for \( m \)**:
   \[
   m = 12 - 7 = 5
   \]

### Final Answer:
\[
m = 5
\]

Would you like more details, or do you have any questions?

---

Here are five related questions for further practice:
1. What if \( 2^m \times 2^6 \times 2^{-4} / 2^{-5} = 2^{10} \)?
2. How would the solution change if the denominator was \( 2^5 \) instead of \( 2^{-5} \)?
3. How do exponent rules change if we are dealing with a base other than 2?
4. Can this type of problem be solved using logarithms? Why or why not?
5. How can we verify the result by plugging \( m = 5 \) back into the original equation?

**Tip:** When combining exponents, remember that multiplication of terms with the same base requires adding exponents, while division involves subtracting them.

Find m if \( \frac{2^m \times 2^6 \times 2^{-4}}{2^{-5}} = 2^{12} \).

Learn how to solve for m in the equation \( \frac{2^m \times 2^6 \times 2^{-4}}{2^{-5}} = 2^{12} \) by applying the laws of exponents. This problem is ideal for students in Grades 9-10, covering key concepts in algebra and exponent rules.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation