Solve {[(27^{3}:9^{3})+3]:[2^{4}:2^{3}+2^3]}+(6^3)^2:6^4

$$\{ [ ( 27 ^ { 3 } : 9 ^ { 3 } ) + 3 ] : [ 2 ^ { 4 } : 2 ^ { 3 } + 2 ^ { 3 } ] \} + ( 6 ^ { 3 } ) ^ { 2 } : 6 ^ { 4 }$$

Evaluate

$$39$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 3 from 4 to get 1.

$$\frac{\frac{27^{3}}{9^{3}}+3}{2^{1}+2^{3}}+\frac{\left(6^{3}\right)^{2}}{6^{4}}$$

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

$$\frac{\frac{27^{3}}{9^{3}}+3}{2^{1}+2^{3}}+\frac{6^{6}}{6^{4}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 4 from 6 to get 2.

$$\frac{\frac{27^{3}}{9^{3}}+3}{2^{1}+2^{3}}+6^{2}$$

Calculate 27 to the power of 3 and get 19683.

$$\frac{\frac{19683}{9^{3}}+3}{2^{1}+2^{3}}+6^{2}$$

Calculate 9 to the power of 3 and get 729.

$$\frac{\frac{19683}{729}+3}{2^{1}+2^{3}}+6^{2}$$

Divide 19683 by 729 to get 27.

$$\frac{27+3}{2^{1}+2^{3}}+6^{2}$$

Add 27 and 3 to get 30.

$$\frac{30}{2^{1}+2^{3}}+6^{2}$$

Calculate 2 to the power of 1 and get 2.

$$\frac{30}{2+2^{3}}+6^{2}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{30}{2+8}+6^{2}$$

Add 2 and 8 to get 10.

$$\frac{30}{10}+6^{2}$$

Divide 30 by 10 to get 3.

$$3+6^{2}$$

Calculate 6 to the power of 2 and get 36.

$$3+36$$

Add 3 and 36 to get 39.

$$39$$

Factor

$$3\times 13$$