Simplify (3^5 * 27^3 * 9^4) / (3 * 81^4), giving the answer as a power of 3 and as a whole number.

Step-by-step solution

1. 1

   Put everything over a single base. The exponent laws only combine powers that share a base, and $27$, $9$ and $81$ are all powers of $3$:

   $27=3^{3},\qquad 9=3^{2},\qquad 81=3^{4}.$

2. 2

   Rewrite each factor using the power-of-a-power rule. Multiply the exponents:

   $27^{3}=\left(3^{3}\right)^{3}=3^{9},\qquad 9^{4}=\left(3^{2}\right)^{4}=3^{8},\qquad 81^{4}=\left(3^{4}\right)^{4}=3^{16}.$

3. 3

   Collect the numerator and the denominator separately. Multiplying like bases adds exponents:

   $\text{numerator}=3^{5}\cdot 3^{9}\cdot 3^{8}=3^{5+9+8}=3^{22},$

   $\text{denominator}=3^{1}\cdot 3^{16}=3^{1+16}=3^{17}.$

   The lone $3$ counts as $3^1$ — forgetting that exponent is the usual slip here.

4. 4

   Divide by subtracting exponents.

   $\frac{3^{22}}{3^{17}}=3^{22-17}=3^{5}.$

5. 5

   Evaluate and verify. $3^{5}=243$. As a check, the raw integers are $3^5\cdot27^3\cdot9^4=243\cdot19683\cdot6561=31\,381\,059\,609$ and $3\cdot81^4=3\cdot43\,046\,721=129\,140\,163$; dividing gives exactly $243$.