Understanding the Exponent Equation

The question asks for the ratio m:n given the equation $$(27)^m = (81)^n$$. To find this ratio, we need to simplify the equation by expressing both bases, 27 and 81, as powers of the same base number. The most suitable common base here is 3, because 27 and 81 are both powers of 3.

Converting Bases to a Common Base (3)

Let's express 27 and 81 in terms of base 3:

• $$27 = 3 \times 3 \times 3 = 3^3$$
• $$81 = 3 \times 3 \times 3 \times 3 = 3^4$$

Substituting into the Given Equation

Now, substitute these equivalent expressions back into the original equation $$(27)^m = (81)^n$$:

$$\left(3^3\right)^m = \left(3^4\right)^n$$

Applying Exponent Rules

We use the rule of exponents that states $$(a^x)^y = a^{xy}$$. Applying this rule to both sides of the equation:

• The left side becomes $$3^{3 \times m} = 3^{3m}$$
• The right side becomes $$3^{4 \times n} = 3^{4n}$$

So the equation is now:

$$3^{3m} = 3^{4n}$$

Solving for the Ratio m:n

When we have an equation where the bases are equal, the exponents must also be equal. Therefore, from $$3^{3m} = 3^{4n}$$, we can conclude:

$$3m = 4n$$

To find the ratio m:n, we can rearrange this equation by dividing both sides by n (assuming n is not zero) and by 3:

$$\frac{3m}{n} = 4$$

$$\frac{m}{n} = \frac{4}{3}$$

The ratio m:n is therefore 4:3.

Final Answer

Based on the calculations, the ratio m:n is 4:3.