Solve frac{3^{3}*(343)^{2/3}*25^{1/2}*a^{-2}*b^{0}*c}{(3a^5b^3c)^2*5^3}

$$\frac { 3 ^ { 3 } \cdot ( 343 ) ^ { 2 / 3 } \cdot 25 ^ { 1 / 2 } \cdot a ^ { - 2 } \cdot b ^ { 0 } \cdot c } { ( 3 a ^ { 5 } b ^ { 3 } c ) ^ { 2 } \cdot 5 ^ { 3 } }$$

Evaluate

$$\frac{147}{25cb^{6}a^{12}}$$

Solution Steps

Calculate 3 to the power of 3 and get 27.

$$\frac{27\times 343^{\frac{2}{3}}\times 25^{\frac{1}{2}}a^{-2}b^{0}c}{\left(3a^{5}b^{3}c\right)^{2}\times 5^{3}}$$

Calculate 343 to the power of \frac{2}{3} and get 49.

$$\frac{27\times 49\times 25^{\frac{1}{2}}a^{-2}b^{0}c}{\left(3a^{5}b^{3}c\right)^{2}\times 5^{3}}$$

Multiply 27 and 49 to get 1323.

$$\frac{1323\times 25^{\frac{1}{2}}a^{-2}b^{0}c}{\left(3a^{5}b^{3}c\right)^{2}\times 5^{3}}$$

Calculate 25 to the power of \frac{1}{2} and get 5.

$$\frac{1323\times 5a^{-2}b^{0}c}{\left(3a^{5}b^{3}c\right)^{2}\times 5^{3}}$$

Multiply 1323 and 5 to get 6615.

$$\frac{6615a^{-2}b^{0}c}{\left(3a^{5}b^{3}c\right)^{2}\times 5^{3}}$$

Calculate b to the power of 0 and get 1.

$$\frac{6615a^{-2}\times 1c}{\left(3a^{5}b^{3}c\right)^{2}\times 5^{3}}$$

Multiply 6615 and 1 to get 6615.

$$\frac{6615a^{-2}c}{\left(3a^{5}b^{3}c\right)^{2}\times 5^{3}}$$

Expand \left(3a^{5}b^{3}c\right)^{2}.

$$\frac{6615a^{-2}c}{3^{2}\left(a^{5}\right)^{2}\left(b^{3}\right)^{2}c^{2}\times 5^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\frac{6615a^{-2}c}{3^{2}a^{10}\left(b^{3}\right)^{2}c^{2}\times 5^{3}}$$

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

$$\frac{6615a^{-2}c}{3^{2}a^{10}b^{6}c^{2}\times 5^{3}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{6615a^{-2}c}{9a^{10}b^{6}c^{2}\times 5^{3}}$$

Calculate 5 to the power of 3 and get 125.

$$\frac{6615a^{-2}c}{9a^{10}b^{6}c^{2}\times 125}$$

Multiply 9 and 125 to get 1125.

$$\frac{6615a^{-2}c}{1125a^{10}b^{6}c^{2}}$$

Cancel out 45c in both numerator and denominator.

$$\frac{147a^{-2}}{25cb^{6}a^{10}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{147}{25cb^{6}a^{12}}$$

Differentiate w.r.t. b

$$-\frac{882}{25cb^{7}a^{12}}$$