Solve for p 3^(p-7)*3^(2p+1)=3^(8p-36)

$3^{p - 7} \cdot 3^{2 p + 1} = 3^{8 p - 36}$

Step 1

Multiply $3^{p - 7}$ by $3^{2 p + 1}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3^{p - 7 + 2 p + 1} = 3^{8 p - 36}$

Add $p$ and $2 p$ .

$3^{3 p - 7 + 1} = 3^{8 p - 36}$

Add $- 7$ and $1$ .

$3^{3 p - 6} = 3^{8 p - 36}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 p - 6 = 8 p - 36$

Step 3

Solve for $p$ .

Move all terms containing $p$ to the left side of the equation.

Subtract $8 p$ from both sides of the equation.

$3 p - 6 - 8 p = - 36$

Subtract $8 p$ from $3 p$ .

$- 5 p - 6 = - 36$

Move all terms not containing $p$ to the right side of the equation.

Add $6$ to both sides of the equation.

$- 5 p = - 36 + 6$

Add $- 36$ and $6$ .

$- 5 p = - 30$

Divide each term in $- 5 p = - 30$ by $- 5$ and simplify.

Divide each term in $- 5 p = - 30$ by $- 5$ .

$\frac{- 5 p}{- 5} = \frac{- 30}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 p}{- 5} = \frac{- 30}{- 5}$

Divide $p$ by $1$ .

$p = \frac{- 30}{- 5}$

Simplify the right side.

Divide $- 30$ by $- 5$ .

$p = 6$