If 3A−B=7, what is the value of $$\frac{27^A}{3^B}$$ A) $$3^{7}$$ B) $$3^3$$ C) $$7^3$$ D) Not enough information given.

Answer 1

The equation 3A−B=7 can be rearranged and written as: 3A=7+B

Next, $$\frac{27^A}{3^B}$$ can be written as $$\frac{3^{3A}}{3^B}$$. This is because the numerator $$27^A$$ is the same as $$(3)^{(3)^A}$$ since 27=$$3^3$$. Because of the Power of a Product Rule, $$(3)^{(3)^A}$$ can be written as $$(3)^{(3A)}$$

Since 3A=7+B as shown above, $$\frac{3^{3A}}{3^B}$$ is equivalent to $$\frac{3^{7+B}}{3^B}$$.

Because of Product of Power rules, $$\frac{3^{7+B}}{3^B}$$ can be adjusted to $$\frac{3^{7}3^B}{3^B}$$. Simplifying this fraction by dividing the top and bottom by $$3^B$$ results in: $$3^{7}$$ which is the answer (choice A).