Solve [3^{4}*3^{2}*3^{3}:3^{5}-3^{0}*(3^{4})^{2}:(3^{2})^3]:[(3^3)^5:3^19]-3

$$[ 3 ^ { 4 } \times 3 ^ { 2 } \times 3 ^ { 3 } : 3 ^ { 5 } - 3 ^ { 0 } \times ( 3 ^ { 4 } ) ^ { 2 } : ( 3 ^ { 2 } ) ^ { 3 } ] : [ ( 3 ^ { 3 } ) ^ { 5 } : 3 ^ { 19 } ] - 3$$

Evaluate

$$5829$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\frac{3^{6}\times 3^{3}}{3^{5}}-\frac{3^{0}\times \left(3^{4}\right)^{2}}{\left(3^{2}\right)^{3}}}{\frac{\left(3^{3}\right)^{5}}{3^{19}}}-3$$

To multiply powers of the same base, add their exponents. Add 6 and 3 to get 9.

$$\frac{\frac{3^{9}}{3^{5}}-\frac{3^{0}\times \left(3^{4}\right)^{2}}{\left(3^{2}\right)^{3}}}{\frac{\left(3^{3}\right)^{5}}{3^{19}}}-3$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 5 from 9 to get 4.

$$\frac{3^{4}-\frac{3^{0}\times \left(3^{4}\right)^{2}}{\left(3^{2}\right)^{3}}}{\frac{\left(3^{3}\right)^{5}}{3^{19}}}-3$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{3^{4}-\frac{3^{0}\times 3^{8}}{\left(3^{2}\right)^{3}}}{\frac{\left(3^{3}\right)^{5}}{3^{19}}}-3$$

To multiply powers of the same base, add their exponents. Add 0 and 8 to get 8.

$$\frac{3^{4}-\frac{3^{8}}{\left(3^{2}\right)^{3}}}{\frac{\left(3^{3}\right)^{5}}{3^{19}}}-3$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{3^{4}-\frac{3^{8}}{3^{6}}}{\frac{\left(3^{3}\right)^{5}}{3^{19}}}-3$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 6 from 8 to get 2.

$$\frac{3^{4}-3^{2}}{\frac{\left(3^{3}\right)^{5}}{3^{19}}}-3$$

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$\frac{3^{4}-3^{2}}{\frac{3^{15}}{3^{19}}}-3$$

Rewrite 3^{19} as 3^{15}\times 3^{4}. Cancel out 3^{15} in both numerator and denominator.

$$\frac{3^{4}-3^{2}}{\frac{1}{3^{4}}}-3$$

Calculate 3 to the power of 4 and get 81.

$$\frac{81-3^{2}}{\frac{1}{3^{4}}}-3$$

Calculate 3 to the power of 2 and get 9.

$$\frac{81-9}{\frac{1}{3^{4}}}-3$$

Subtract 9 from 81 to get 72.

$$\frac{72}{\frac{1}{3^{4}}}-3$$

Calculate 3 to the power of 4 and get 81.

$$\frac{72}{\frac{1}{81}}-3$$

Divide 72 by \frac{1}{81} by multiplying 72 by the reciprocal of \frac{1}{81}.

$$72\times 81-3$$

Multiply 72 and 81 to get 5832.

$$5832-3$$

Subtract 3 from 5832 to get 5829.

$$5829$$

Factor

$$3\times 29\times 67$$