B4 Simplify the expression $3^{2 n-1} \cdot 4^{n+1}+9^{n+1} \cdot 2^{2 n-1}+6^{2 n-1}$ and write it in the form of a power with base 6.

(7 points)

Space for solving tasks of set $B$.

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-06.jpg?height=83&width=449&top_left_y=200&top_left_x=1456)

## PROBLEMS FOR THIRD YEAR STUDENTS

Time for solving: 90 minutes. In set A, a correct answer will be scored 2 points, while an incorrect answer will result in a deduction of half a point. Write the answers of set A in the left table.
![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-06.jpg?height=168&width=1194&top_left_y=578&top_left_x=448)
B4. The expression is converted to powers with the same base $3^{2 n-1} \cdot\left(2^{2}\right)^{n+1}+\left(3^{2}\right)^{n+1} \cdot 2^{2 n-1}+(3 \cdot 2)^{2 n-1}$, and organized as $3^{2 n-1} \cdot 2^{2(n+1)}+3^{2(n+1)} \cdot 2^{2 n-1}+3^{2 n-1} \cdot 2^{2 n-1}$. We factor out the common factor $3^{2 n-1} \cdot 2^{n+1}$ and get $3^{2 n-1} \cdot 2^{n+1}\left(2^{3}+3^{3}+1\right)=6^{2 n-1} \cdot 36$. The number 36 is written as a power of $6^{2}$ and organized as $6^{2 n-1} \cdot 6^{2}=6^{2 n+1}$.

Reformulating the first term $3^{2 n-1} \cdot 4^{n+1}=3^{2 n-1} \cdot 2^{2(n+1)} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point

Reformulating the second term $9^{n+1} \cdot 2^{2 n-1}=3^{2(n+1)} \cdot 2^{2 n-1} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point

Reformulating the third term $6^{2 n-1}=(3 \cdot 2)^{2 n-1}=3^{2 n-1} \cdot 2^{2 n-1} \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-13.jpg?height=57&width=1685&top_left_y=2459&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-13.jpg?height=57&width=1685&top_left_y=2507&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-13.jpg?height=51&width=1685&top_left_y=2556&top_left_x=228)

## Third Year

Group A

| $\mathrm{A} 1$ | $\mathrm{~A} 2$ | $\mathrm{~A} 3$ | $\mathrm{~A} 4$ | $\mathrm{~A} 5$ | $\mathrm{~A} 6$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathrm{A}$ | $\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{A}$ | $\mathrm{D}$ | $\mathrm{C}$ |