Problem 89

Given $a^{m}=3$ and $a^{n}=2$, we are asked to find $a^{2m+3n}$. Let's break this down step by step:

1. **Express $a^{2m+3n}$ in terms of $a^{m}$ and $a^{n}$:**
$$a^{2m+3n} = a^{2m} \cdot a^{3n}$$
This step uses the property of exponents that $a^{b+c} = a^b \cdot a^c$.

2. **Substitute $a^{m}$ and $a^{n}$ with their given values:**
$$a^{2m} \cdot a^{3n} = (a^{m})^{2} \cdot (a^{n})^{3}$$
Here, we use the property that $(a^b)^c = a^{bc}$.

3. **Replace $a^{m}$ and $a^{n}$ with their respective values:**
$$(a^{m})^{2} \cdot (a^{n})^{3} = 3^{2} \times 2^{3}$$
Since $a^{m} = 3$ and $a^{n} = 2$, we replace them directly.

4. Calculate the powers and multiply:
$$3^{2} \times 2^{3} = 9 \times 8$$
Here, we calculate $3^2 = 9$ and $2^3 = 8$.

5. Multiply the results to find the final answer:
$$9 \times 8 = 72$$
Multiplying the two results gives us the final answer.

Therefore, $a^{2m+3n} = 72$. So, the answer is $\boxed{72}$.