Solve (frac{9c^{5}}{a^{3}b^{-2}})^{-2}:(a^2b^-3/6c^4)^3

$$( \frac { 9 c ^ { 5 } } { a ^ { 3 } b ^ { - 2 } } ) ^ { - 2 } : ( \frac { a ^ { 2 } b ^ { - 3 } } { 6 c ^ { 4 } } ) ^ { 3 }$$

Evaluate

$$\frac{8c^{2}b^{5}}{3}$$

Solution Steps

To raise \frac{9c^{5}}{a^{3}b^{-2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(9c^{5}\right)^{-2}}{\left(a^{3}b^{-2}\right)^{-2}}}{\left(\frac{a^{2}b^{-3}}{6c^{4}}\right)^{3}}$$

To raise \frac{a^{2}b^{-3}}{6c^{4}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(9c^{5}\right)^{-2}}{\left(a^{3}b^{-2}\right)^{-2}}}{\frac{\left(a^{2}b^{-3}\right)^{3}}{\left(6c^{4}\right)^{3}}}$$

Divide \frac{\left(9c^{5}\right)^{-2}}{\left(a^{3}b^{-2}\right)^{-2}} by \frac{\left(a^{2}b^{-3}\right)^{3}}{\left(6c^{4}\right)^{3}} by multiplying \frac{\left(9c^{5}\right)^{-2}}{\left(a^{3}b^{-2}\right)^{-2}} by the reciprocal of \frac{\left(a^{2}b^{-3}\right)^{3}}{\left(6c^{4}\right)^{3}}.

$$\frac{\left(9c^{5}\right)^{-2}\times \left(6c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Expand \left(9c^{5}\right)^{-2}.

$$\frac{9^{-2}\left(c^{5}\right)^{-2}\times \left(6c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{9^{-2}c^{-10}\times \left(6c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Calculate 9 to the power of -2 and get \frac{1}{81}.

$$\frac{\frac{1}{81}c^{-10}\times \left(6c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Expand \left(6c^{4}\right)^{3}.

$$\frac{\frac{1}{81}c^{-10}\times 6^{3}\left(c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{\frac{1}{81}c^{-10}\times 6^{3}c^{12}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Calculate 6 to the power of 3 and get 216.

$$\frac{\frac{1}{81}c^{-10}\times 216c^{12}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Multiply \frac{1}{81} and 216 to get \frac{8}{3}.

$$\frac{\frac{8}{3}c^{-10}c^{12}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -10 and 12 to get 2.

$$\frac{\frac{8}{3}c^{2}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{\left(a^{3}\right)^{-2}\left(b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{a^{-6}\left(b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{a^{-6}b^{4}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{a^{-6}b^{4}\left(a^{2}\right)^{3}\left(b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{a^{-6}b^{4}a^{6}\left(b^{-3}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{\frac{8}{3}c^{2}}{a^{-6}b^{4}a^{6}b^{-9}}$$

Multiply a^{-6} and a^{6} to get 1.

$$\frac{\frac{8}{3}c^{2}}{b^{4}b^{-9}}$$

To multiply powers of the same base, add their exponents. Add 4 and -9 to get -5.

$$\frac{\frac{8}{3}c^{2}}{b^{-5}}$$

Expand

$$\frac{8c^{2}b^{5}}{3}$$

Solution Steps

To raise \frac{9c^{5}}{a^{3}b^{-2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(9c^{5}\right)^{-2}}{\left(a^{3}b^{-2}\right)^{-2}}}{\left(\frac{a^{2}b^{-3}}{6c^{4}}\right)^{3}}$$

To raise \frac{a^{2}b^{-3}}{6c^{4}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(9c^{5}\right)^{-2}}{\left(a^{3}b^{-2}\right)^{-2}}}{\frac{\left(a^{2}b^{-3}\right)^{3}}{\left(6c^{4}\right)^{3}}}$$

Divide \frac{\left(9c^{5}\right)^{-2}}{\left(a^{3}b^{-2}\right)^{-2}} by \frac{\left(a^{2}b^{-3}\right)^{3}}{\left(6c^{4}\right)^{3}} by multiplying \frac{\left(9c^{5}\right)^{-2}}{\left(a^{3}b^{-2}\right)^{-2}} by the reciprocal of \frac{\left(a^{2}b^{-3}\right)^{3}}{\left(6c^{4}\right)^{3}}.

$$\frac{\left(9c^{5}\right)^{-2}\times \left(6c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Expand \left(9c^{5}\right)^{-2}.

$$\frac{9^{-2}\left(c^{5}\right)^{-2}\times \left(6c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{9^{-2}c^{-10}\times \left(6c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Calculate 9 to the power of -2 and get \frac{1}{81}.

$$\frac{\frac{1}{81}c^{-10}\times \left(6c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Expand \left(6c^{4}\right)^{3}.

$$\frac{\frac{1}{81}c^{-10}\times 6^{3}\left(c^{4}\right)^{3}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{\frac{1}{81}c^{-10}\times 6^{3}c^{12}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Calculate 6 to the power of 3 and get 216.

$$\frac{\frac{1}{81}c^{-10}\times 216c^{12}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

Multiply \frac{1}{81} and 216 to get \frac{8}{3}.

$$\frac{\frac{8}{3}c^{-10}c^{12}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -10 and 12 to get 2.

$$\frac{\frac{8}{3}c^{2}}{\left(a^{3}b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{\left(a^{3}\right)^{-2}\left(b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{a^{-6}\left(b^{-2}\right)^{-2}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{a^{-6}b^{4}\left(a^{2}b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{a^{-6}b^{4}\left(a^{2}\right)^{3}\left(b^{-3}\right)^{3}}$$

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

$$\frac{\frac{8}{3}c^{2}}{a^{-6}b^{4}a^{6}\left(b^{-3}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{\frac{8}{3}c^{2}}{a^{-6}b^{4}a^{6}b^{-9}}$$

Multiply a^{-6} and a^{6} to get 1.

$$\frac{\frac{8}{3}c^{2}}{b^{4}b^{-9}}$$

To multiply powers of the same base, add their exponents. Add 4 and -9 to get -5.

$$\frac{\frac{8}{3}c^{2}}{b^{-5}}$$