Solve frac{(4a^{2}b^{-3})^{2}(b^{-7}c^{8})^{-3}}{(2b^{3}c^{-5})^3(a^-5b^-6)^-4}

$$\frac { ( 4 a ^ { 2 } b ^ { - 3 } ) ^ { 2 } ( b ^ { - 7 } c ^ { 8 } ) ^ { - 3 } } { ( 2 b ^ { 3 } c ^ { - 5 } ) ^ { 3 } ( a ^ { - 5 } b ^ { - 6 } ) ^ { - 4 } }$$

Evaluate

$$\frac{2}{c^{9}a^{16}b^{18}}$$

Solution Steps

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

$$\frac{4^{2}\left(a^{2}\right)^{2}\left(b^{-3}\right)^{2}\left(b^{-7}c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{4^{2}a^{4}\left(b^{-3}\right)^{2}\left(b^{-7}c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{4^{2}a^{4}b^{-6}\left(b^{-7}c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16a^{4}b^{-6}\left(b^{-7}c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

Expand \left(b^{-7}c^{8}\right)^{-3}.

$$\frac{16a^{4}b^{-6}\left(b^{-7}\right)^{-3}\left(c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -7 and -3 to get 21.

$$\frac{16a^{4}b^{-6}b^{21}\left(c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{16a^{4}b^{-6}b^{21}c^{-24}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

To multiply powers of the same base, add their exponents. Add -6 and 21 to get 15.

$$\frac{16a^{4}b^{15}c^{-24}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{16a^{4}b^{15}c^{-24}}{2^{3}\left(b^{3}\right)^{3}\left(c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{16a^{4}b^{15}c^{-24}}{2^{3}b^{9}\left(c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and 3 to get -15.

$$\frac{16a^{4}b^{15}c^{-24}}{2^{3}b^{9}c^{-15}\left(a^{-5}b^{-6}\right)^{-4}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{9}c^{-15}\left(a^{-5}b^{-6}\right)^{-4}}$$

Expand \left(a^{-5}b^{-6}\right)^{-4}.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{9}c^{-15}\left(a^{-5}\right)^{-4}\left(b^{-6}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -4 to get 20.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{9}c^{-15}a^{20}\left(b^{-6}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -6 and -4 to get 24.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{9}c^{-15}a^{20}b^{24}}$$

To multiply powers of the same base, add their exponents. Add 9 and 24 to get 33.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{33}c^{-15}a^{20}}$$

Cancel out 8a^{4}b^{15} in both numerator and denominator.

$$\frac{2c^{-24}}{c^{-15}a^{16}b^{18}}$$

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

$$\frac{2}{c^{9}a^{16}b^{18}}$$

Expand

$$\frac{2}{c^{9}a^{16}b^{18}}$$

Solution Steps

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

$$\frac{4^{2}\left(a^{2}\right)^{2}\left(b^{-3}\right)^{2}\left(b^{-7}c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{4^{2}a^{4}\left(b^{-3}\right)^{2}\left(b^{-7}c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{4^{2}a^{4}b^{-6}\left(b^{-7}c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16a^{4}b^{-6}\left(b^{-7}c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

Expand \left(b^{-7}c^{8}\right)^{-3}.

$$\frac{16a^{4}b^{-6}\left(b^{-7}\right)^{-3}\left(c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -7 and -3 to get 21.

$$\frac{16a^{4}b^{-6}b^{21}\left(c^{8}\right)^{-3}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{16a^{4}b^{-6}b^{21}c^{-24}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

To multiply powers of the same base, add their exponents. Add -6 and 21 to get 15.

$$\frac{16a^{4}b^{15}c^{-24}}{\left(2b^{3}c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{16a^{4}b^{15}c^{-24}}{2^{3}\left(b^{3}\right)^{3}\left(c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

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

$$\frac{16a^{4}b^{15}c^{-24}}{2^{3}b^{9}\left(c^{-5}\right)^{3}\left(a^{-5}b^{-6}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and 3 to get -15.

$$\frac{16a^{4}b^{15}c^{-24}}{2^{3}b^{9}c^{-15}\left(a^{-5}b^{-6}\right)^{-4}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{9}c^{-15}\left(a^{-5}b^{-6}\right)^{-4}}$$

Expand \left(a^{-5}b^{-6}\right)^{-4}.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{9}c^{-15}\left(a^{-5}\right)^{-4}\left(b^{-6}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -4 to get 20.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{9}c^{-15}a^{20}\left(b^{-6}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -6 and -4 to get 24.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{9}c^{-15}a^{20}b^{24}}$$

To multiply powers of the same base, add their exponents. Add 9 and 24 to get 33.

$$\frac{16a^{4}b^{15}c^{-24}}{8b^{33}c^{-15}a^{20}}$$

Cancel out 8a^{4}b^{15} in both numerator and denominator.

$$\frac{2c^{-24}}{c^{-15}a^{16}b^{18}}$$

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

$$\frac{2}{c^{9}a^{16}b^{18}}$$