12. $$ \frac{81^{3 m+2}}{243^{m}}=27 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{81^{3m + 2}}{243^{m}} = 27
$$

**Step 1: Express All Numbers as Powers of 3**

First, recognize that all numbers in the equation can be expressed as powers of 3:

- $$81 = 3^4$$
- $$243 = 3^5$$
- $$27 = 3^3$$

**Step 2: Substitute These Values into the Equation**

Replace each number with its corresponding power of 3:

$$
\frac{(3^4)^{3m + 2}}{(3^5)^m} = 3^3
$$

**Step 3: Simplify the Exponents**

Apply the power of a power rule $$(a^b)^c = a^{b \cdot c}$$:

$$
\frac{3^{4(3m + 2)}}{3^{5m}} = 3^3
$$

Calculate the exponents:

$$
\frac{3^{12m + 8}}{3^{5m}} = 3^3
$$

**Step 4: Use the Quotient Rule for Exponents**

The quotient rule states that $$\frac{a^b}{a^c} = a^{b - c}$$:

$$
3^{12m + 8 - 5m} = 3^3
$$

Simplify the exponent:

$$
3^{7m + 8} = 3^3
$$

**Step 5: Set the Exponents Equal to Each Other**

Since the bases are the same and non-zero, their exponents must be equal:

$$
7m + 8 = 3
$$

**Step 6: Solve for $$m$$**

Subtract 8 from both sides:

$$
7m = 3 - 8
$$
$$
7m = -5
$$

Divide both sides by 7:

$$
m = \frac{-5}{7}
$$

**Final Answer:**

$$
m = -\frac{5}{7}
$$
$$ m = -\frac{5}{7} $$

12. \( \frac{81^{3 m+2}}{243^{m}}=27 \)