Solve =[frac{(a^{2}b^{3/4}c^{1/2})^{4}}{(a^{1/3}b^frac{1{6}}c^frac{1{2}})^-6}]^-1

$$= [ \frac { ( a ^ { 2 } b ^ { \frac { 3 } { 4 } } c ^ { \frac { 1 } { 2 } } ) ^ { 4 } } { ( a ^ { \frac { 1 } { 3 } } b ^ { \frac { 1 } { 6 } } c ^ { \frac { 1 } { 2 } } ) ^ { - 6 } } ] ^ { - 1 }$$

Evaluate

$$\frac{1}{b^{4}c^{5}a^{10}}$$

Solution Steps

Expand \left(a^{2}b^{\frac{3}{4}}c^{\frac{1}{2}}\right)^{4}.

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

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

$$\left(\frac{a^{8}\left(b^{\frac{3}{4}}\right)^{4}\left(c^{\frac{1}{2}}\right)^{4}}{\left(a^{\frac{1}{3}}b^{\frac{1}{6}}c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{4} and 4 to get 3.

$$\left(\frac{a^{8}b^{3}\left(c^{\frac{1}{2}}\right)^{4}}{\left(a^{\frac{1}{3}}b^{\frac{1}{6}}c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and 4 to get 2.

$$\left(\frac{a^{8}b^{3}c^{2}}{\left(a^{\frac{1}{3}}b^{\frac{1}{6}}c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

Expand \left(a^{\frac{1}{3}}b^{\frac{1}{6}}c^{\frac{1}{2}}\right)^{-6}.

$$\left(\frac{a^{8}b^{3}c^{2}}{\left(a^{\frac{1}{3}}\right)^{-6}\left(b^{\frac{1}{6}}\right)^{-6}\left(c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and -6 to get -2.

$$\left(\frac{a^{8}b^{3}c^{2}}{a^{-2}\left(b^{\frac{1}{6}}\right)^{-6}\left(c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{6} and -6 to get -1.

$$\left(\frac{a^{8}b^{3}c^{2}}{a^{-2}b^{-1}\left(c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and -6 to get -3.

$$\left(\frac{a^{8}b^{3}c^{2}}{a^{-2}b^{-1}c^{-3}}\right)^{-1}$$

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

$$\left(b^{4}c^{5}a^{10}\right)^{-1}$$

Expand \left(b^{4}c^{5}a^{10}\right)^{-1}.

$$\left(b^{4}\right)^{-1}\left(c^{5}\right)^{-1}\left(a^{10}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

$$b^{-4}\left(c^{5}\right)^{-1}\left(a^{10}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -1 to get -5.

$$b^{-4}c^{-5}\left(a^{10}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply 10 and -1 to get -10.

$$b^{-4}c^{-5}a^{-10}$$

Expand

$$\frac{1}{b^{4}c^{5}a^{10}}$$

Solution Steps

Expand \left(a^{2}b^{\frac{3}{4}}c^{\frac{1}{2}}\right)^{4}.

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

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

$$\left(\frac{a^{8}\left(b^{\frac{3}{4}}\right)^{4}\left(c^{\frac{1}{2}}\right)^{4}}{\left(a^{\frac{1}{3}}b^{\frac{1}{6}}c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{4} and 4 to get 3.

$$\left(\frac{a^{8}b^{3}\left(c^{\frac{1}{2}}\right)^{4}}{\left(a^{\frac{1}{3}}b^{\frac{1}{6}}c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and 4 to get 2.

$$\left(\frac{a^{8}b^{3}c^{2}}{\left(a^{\frac{1}{3}}b^{\frac{1}{6}}c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

Expand \left(a^{\frac{1}{3}}b^{\frac{1}{6}}c^{\frac{1}{2}}\right)^{-6}.

$$\left(\frac{a^{8}b^{3}c^{2}}{\left(a^{\frac{1}{3}}\right)^{-6}\left(b^{\frac{1}{6}}\right)^{-6}\left(c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and -6 to get -2.

$$\left(\frac{a^{8}b^{3}c^{2}}{a^{-2}\left(b^{\frac{1}{6}}\right)^{-6}\left(c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{6} and -6 to get -1.

$$\left(\frac{a^{8}b^{3}c^{2}}{a^{-2}b^{-1}\left(c^{\frac{1}{2}}\right)^{-6}}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and -6 to get -3.

$$\left(\frac{a^{8}b^{3}c^{2}}{a^{-2}b^{-1}c^{-3}}\right)^{-1}$$

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

$$\left(b^{4}c^{5}a^{10}\right)^{-1}$$

Expand \left(b^{4}c^{5}a^{10}\right)^{-1}.

$$\left(b^{4}\right)^{-1}\left(c^{5}\right)^{-1}\left(a^{10}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

$$b^{-4}\left(c^{5}\right)^{-1}\left(a^{10}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -1 to get -5.

$$b^{-4}c^{-5}\left(a^{10}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply 10 and -1 to get -10.

$$b^{-4}c^{-5}a^{-10}$$