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Let's solve the given equations step by step to find the value of \( m \).

### (a) \( \frac{2^m}{2^{-2}} = 2^3 \)

Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we simplify:

\[
\frac{2^m}{2^{-2}} = 2^{m - (-2)} = 2^{m + 2}
\]

Thus, the equation becomes:

\[
2^{m + 2} = 2^3
\]

Since the bases are the same, equate the exponents:

\[
m + 2 = 3
\]

Solve for \( m \):

\[
m = 1
\]

---

### (b) \( \frac{3^m \times 3^6}{3^{-3}} = 3^{18} \)

Using the property \( a^m \times a^n = a^{m+n} \) for the numerator and simplifying:

\[
\frac{3^m \times 3^6}{3^{-3}} = 3^{m + 6 - (-3)} = 3^{m + 6 + 3} = 3^{m + 9}
\]

Thus, the equation becomes:

\[
3^{m + 9} = 3^{18}
\]

Since the bases are the same, equate the exponents:

\[
m + 9 = 18
\]

Solve for \( m \):

\[
m = 9
\]

---

### (c) \( \frac{6^m \times 6^3 \times 6^{-2}}{6^{-5}} = 6^{12} \)

Simplifying the numerator and denominator separately using the exponent rules:

\[
6^m \times 6^3 \times 6^{-2} = 6^{m + 3 - 2} = 6^{m + 1}
\]

The denominator simplifies as:

\[
6^{-5} = 6^5
\]

So the equation becomes:

\[
\frac{6^{m + 1}}{6^5} = 6^{m + 1 - 5} = 6^{m - 4}
\]

Thus, the equation becomes:

\[
6^{m - 4} = 6^{12}
\]

Since the bases are the same, equate the exponents:

\[
m - 4 = 12
\]

Solve for \( m \):

\[
m = 16
\]

---

### Final answers:
- (a) \( m = 1 \)
- (b) \( m = 9 \)
- (c) \( m = 16 \)

Would you like any further explanation or details? Here are some related questions for practice:

1. Solve for \( m \) in \( \frac{2^{m+1}}{2^3} = 2^4 \).
2. Simplify \( \frac{5^m \times 5^2}{5^{-3}} \).
3. Solve \( \frac{4^m}{4^{-1}} = 4^5 \).
4. Simplify \( 3^{m+2} \times 3^{-4} = 3^6 \).
5. Solve for \( m \) in \( \frac{7^m \times 7^{-3}}{7^4} = 7^{-2} \).

**Tip:** Always remember the basic exponent rules like \( a^m \times a^n = a^{m+n} \) and \( \frac{a^m}{a^n} = a^{m-n} \), as they are extremely helpful for simplifying complex expressions!

Find the value of m in the following equations:
(a) (2^m) / (2^-2) = 2^3
(b) (3^m * 3^6) / (3^-3) = 3^18
(c) (6^m * 6^3 * 6^-2) / (6^-5) = 6^12

Learn how to solve equations involving exponents by finding the value of m. This problem set covers exponentiation rules, properties of exponents, and the equality of exponents. Suitable for students in Grades 8-10.

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