Here’s the solution to the problems provided in the image:

Part (a):

The problem is: $\frac{(3^2) \cdot 5^{-2}}{15^{-1}}$

Step 1: Simplify $3^2$ and $5^{-2}$:

$3^2 = 9,\quad 5^{-2} = \frac{1}{5^2} = \frac{1}{25}.$

The numerator becomes: $9 \cdot \frac{1}{25} = \frac{9}{25}.$

Step 2: Simplify $15^{-1}$:

$15^{-1} = \frac{1}{15}.$

Step 3: Divide the numerator by the denominator:

$\frac{\frac{9}{25}}{\frac{1}{15}} = \frac{9}{25} \cdot 15 = \frac{9 \cdot 15}{25}.$

Simplify: $\frac{135}{25} = \frac{27}{5}.$

Final Answer:

$\frac{27}{5}.$

Part (d):

The problem is: $3^{x-2} = 27.$

Step 1: Rewrite $27$ as a power of $3$:

$27 = 3^3.$

The equation becomes: $3^{x-2} = 3^3.$

Step 2: Set the exponents equal:

$x - 2 = 3.$

Solve for $x$: $x = 5.$

Final Answer:

$x = 5.$

Part (e):

The problem is: $\frac{g^{x+3}}{2} = \frac{1}{8}.$

Step 1: Multiply through by $2$:

$g^{x+3} = \frac{1}{8} \cdot 2 = \frac{1}{4}.$

Step 2: Rewrite $\frac{1}{4}$ as a power of $g$:

Assume $g = 2$, then: $g^{x+3} = \frac{1}{4} = 2^{-2}.$

Step 3: Set the exponents equal:

$x + 3 = -2.$

Solve for $x$: $x = -5.$

Final Answer:

$x = -5.$

Let me know if you would like further clarification or additional details.

Follow-up Questions:

1. How can you generalize solving equations with negative exponents?
2. What steps are key when simplifying fractions with exponents?
3. How do you interpret fractional powers (e.g., $x^{1/2}$) in equations?
4. How would the solution change in part (e) if $g \neq 2$?
5. What methods can you use to verify your solutions?

Tip:

When solving exponential equations, always rewrite both sides with the same base to simplify the comparison of exponents.