Solve {[2n^{3}*25^{3}:(n^{2})^3]*3^6}:(15^2*15^3)=

$$\{ [ 2 n ^ { 3 } \cdot 25 ^ { 3 } : ( n ^ { 2 } ) ^ { 3 } ] \cdot 3 ^ { 6 } \} : ( 15 ^ { 2 } \cdot 15 ^ { 3 } ) =$$

Evaluate

$$\frac{30}{n^{3}}$$

Solution Steps

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

$$\frac{\frac{2n^{3}\times 25^{3}}{n^{6}}\times 3^{6}}{15^{2}\times 15^{3}}$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{\frac{2n^{3}\times 25^{3}}{n^{6}}\times 3^{6}}{15^{5}}$$

Cancel out n^{3} in both numerator and denominator.

$$\frac{\frac{2\times 25^{3}}{n^{3}}\times 3^{6}}{15^{5}}$$

Calculate 25 to the power of 3 and get 15625.

$$\frac{\frac{2\times 15625}{n^{3}}\times 3^{6}}{15^{5}}$$

Multiply 2 and 15625 to get 31250.

$$\frac{\frac{31250}{n^{3}}\times 3^{6}}{15^{5}}$$

Calculate 3 to the power of 6 and get 729.

$$\frac{\frac{31250}{n^{3}}\times 729}{15^{5}}$$

Express \frac{31250}{n^{3}}\times 729 as a single fraction.

$$\frac{\frac{31250\times 729}{n^{3}}}{15^{5}}$$

Calculate 15 to the power of 5 and get 759375.

$$\frac{\frac{31250\times 729}{n^{3}}}{759375}$$

Express \frac{\frac{31250\times 729}{n^{3}}}{759375} as a single fraction.

$$\frac{31250\times 729}{n^{3}\times 759375}$$

Cancel out 243\times 3125 in both numerator and denominator.

$$\frac{3\times 10}{n^{3}}$$

Multiply 3 and 10 to get 30.

$$\frac{30}{n^{3}}$$

Differentiate w.r.t. n

$$-\frac{90}{n^{4}}$$