Solve frac{(2a^{3}b^{-5}c^{4})^{-2}}{(3a^-3b^4c^-8)^3}=

$$\frac { ( 2 a ^ { 3 } b ^ { - 5 } c ^ { 4 } ) ^ { - 2 } } { ( 3 a ^ { - 3 } b ^ { 4 } c ^ { - 8 } ) ^ { 3 } } =$$

Evaluate

$$\frac{a^{3}c^{16}}{108b^{2}}$$

Solution Steps

Expand \left(2a^{3}b^{-5}c^{4}\right)^{-2}.

$$\frac{2^{-2}\left(a^{3}\right)^{-2}\left(b^{-5}\right)^{-2}\left(c^{4}\right)^{-2}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

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

$$\frac{2^{-2}a^{-6}\left(b^{-5}\right)^{-2}\left(c^{4}\right)^{-2}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

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

$$\frac{2^{-2}a^{-6}b^{10}\left(c^{4}\right)^{-2}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

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

$$\frac{2^{-2}a^{-6}b^{10}c^{-8}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

Expand \left(3a^{-3}b^{4}c^{-8}\right)^{3}.

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{3^{3}\left(a^{-3}\right)^{3}\left(b^{4}\right)^{3}\left(c^{-8}\right)^{3}}$$

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

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{3^{3}a^{-9}\left(b^{4}\right)^{3}\left(c^{-8}\right)^{3}}$$

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

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{3^{3}a^{-9}b^{12}\left(c^{-8}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -8 and 3 to get -24.

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{3^{3}a^{-9}b^{12}c^{-24}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{27a^{-9}b^{12}c^{-24}}$$

Cancel out b^{10} in both numerator and denominator.

$$\frac{\frac{1}{4}c^{-8}a^{-6}}{27c^{-24}a^{-9}b^{2}}$$

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

$$\frac{\frac{1}{4}a^{3}c^{16}}{27b^{2}}$$

Expand

$$\frac{a^{3}c^{16}}{108b^{2}}$$

Solution Steps

Expand \left(2a^{3}b^{-5}c^{4}\right)^{-2}.

$$\frac{2^{-2}\left(a^{3}\right)^{-2}\left(b^{-5}\right)^{-2}\left(c^{4}\right)^{-2}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

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

$$\frac{2^{-2}a^{-6}\left(b^{-5}\right)^{-2}\left(c^{4}\right)^{-2}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

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

$$\frac{2^{-2}a^{-6}b^{10}\left(c^{4}\right)^{-2}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

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

$$\frac{2^{-2}a^{-6}b^{10}c^{-8}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{\left(3a^{-3}b^{4}c^{-8}\right)^{3}}$$

Expand \left(3a^{-3}b^{4}c^{-8}\right)^{3}.

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{3^{3}\left(a^{-3}\right)^{3}\left(b^{4}\right)^{3}\left(c^{-8}\right)^{3}}$$

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

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{3^{3}a^{-9}\left(b^{4}\right)^{3}\left(c^{-8}\right)^{3}}$$

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

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{3^{3}a^{-9}b^{12}\left(c^{-8}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -8 and 3 to get -24.

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{3^{3}a^{-9}b^{12}c^{-24}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{\frac{1}{4}a^{-6}b^{10}c^{-8}}{27a^{-9}b^{12}c^{-24}}$$

Cancel out b^{10} in both numerator and denominator.

$$\frac{\frac{1}{4}c^{-8}a^{-6}}{27c^{-24}a^{-9}b^{2}}$$

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

$$\frac{\frac{1}{4}a^{3}c^{16}}{27b^{2}}$$