Solve left(3a^2b^3c^-1right)^2/{left(2{a^{2}b{c}^{0}right)}^{3}}divfrac{{left(2{a}^{3}{b}^2{c}^5right)}^3}{3ab{c}^3}

$$\frac{ { \left(3 { a }^{ 2 } { b }^{ 3 } { c }^{ -1 } \right) }^{ 2 } }{ { \left(2 { a }^{ 2 } b { c }^{ 0 } \right) }^{ 3 } } \div \frac{ { \left(2 { a }^{ 3 } { b }^{ 2 } { c }^{ 5 } \right) }^{ 3 } }{ 3ab { c }^{ 3 } }$$

Evaluate

$$\frac{27}{64b^{2}a^{10}c^{14}}$$

Solution Steps

Divide \frac{\left(3a^{2}b^{3}c^{-1}\right)^{2}}{\left(2a^{2}bc^{0}\right)^{3}} by \frac{\left(2a^{3}b^{2}c^{5}\right)^{3}}{3abc^{3}} by multiplying \frac{\left(3a^{2}b^{3}c^{-1}\right)^{2}}{\left(2a^{2}bc^{0}\right)^{3}} by the reciprocal of \frac{\left(2a^{3}b^{2}c^{5}\right)^{3}}{3abc^{3}}.

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

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

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

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

$$\frac{3^{2}a^{4}\left(b^{3}\right)^{2}\left(c^{-1}\right)^{2}\times 3abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

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

$$\frac{3^{2}a^{4}b^{6}\left(c^{-1}\right)^{2}\times 3abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{3^{2}a^{4}b^{6}c^{-2}\times 3abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9a^{4}b^{6}c^{-2}\times 3abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Multiply 9 and 3 to get 27.

$$\frac{27a^{4}b^{6}c^{-2}abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{27a^{5}b^{6}c^{-2}bc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$\frac{27a^{5}b^{7}c^{-2}c^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -2 and 3 to get 1.

$$\frac{27a^{5}b^{7}c^{1}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Calculate c to the power of 0 and get 1.

$$\frac{27a^{5}b^{7}c^{1}}{\left(2a^{2}b\times 1\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Multiply 2 and 1 to get 2.

$$\frac{27a^{5}b^{7}c^{1}}{\left(2a^{2}b\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{2^{3}\left(a^{2}\right)^{3}b^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{2^{3}a^{6}b^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 2^{3}\left(a^{3}\right)^{3}\left(b^{2}\right)^{3}\left(c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 2^{3}a^{9}\left(b^{2}\right)^{3}\left(c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 2^{3}a^{9}b^{6}\left(c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 2^{3}a^{9}b^{6}c^{15}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 8a^{9}b^{6}c^{15}}$$

Multiply 8 and 8 to get 64.

$$\frac{27a^{5}b^{7}c^{1}}{64a^{6}b^{3}a^{9}b^{6}c^{15}}$$

To multiply powers of the same base, add their exponents. Add 6 and 9 to get 15.

$$\frac{27a^{5}b^{7}c^{1}}{64a^{15}b^{3}b^{6}c^{15}}$$

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\frac{27a^{5}b^{7}c^{1}}{64a^{15}b^{9}c^{15}}$$

Cancel out ca^{5}b^{7} in both numerator and denominator.

$$\frac{27}{64b^{2}a^{10}c^{14}}$$

Expand

$$\frac{27}{64b^{2}a^{10}c^{14}}$$

Solution Steps

Divide \frac{\left(3a^{2}b^{3}c^{-1}\right)^{2}}{\left(2a^{2}bc^{0}\right)^{3}} by \frac{\left(2a^{3}b^{2}c^{5}\right)^{3}}{3abc^{3}} by multiplying \frac{\left(3a^{2}b^{3}c^{-1}\right)^{2}}{\left(2a^{2}bc^{0}\right)^{3}} by the reciprocal of \frac{\left(2a^{3}b^{2}c^{5}\right)^{3}}{3abc^{3}}.

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

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

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

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

$$\frac{3^{2}a^{4}\left(b^{3}\right)^{2}\left(c^{-1}\right)^{2}\times 3abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

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

$$\frac{3^{2}a^{4}b^{6}\left(c^{-1}\right)^{2}\times 3abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{3^{2}a^{4}b^{6}c^{-2}\times 3abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9a^{4}b^{6}c^{-2}\times 3abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Multiply 9 and 3 to get 27.

$$\frac{27a^{4}b^{6}c^{-2}abc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{27a^{5}b^{6}c^{-2}bc^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$\frac{27a^{5}b^{7}c^{-2}c^{3}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -2 and 3 to get 1.

$$\frac{27a^{5}b^{7}c^{1}}{\left(2a^{2}bc^{0}\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Calculate c to the power of 0 and get 1.

$$\frac{27a^{5}b^{7}c^{1}}{\left(2a^{2}b\times 1\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Multiply 2 and 1 to get 2.

$$\frac{27a^{5}b^{7}c^{1}}{\left(2a^{2}b\right)^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{2^{3}\left(a^{2}\right)^{3}b^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{2^{3}a^{6}b^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times \left(2a^{3}b^{2}c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 2^{3}\left(a^{3}\right)^{3}\left(b^{2}\right)^{3}\left(c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 2^{3}a^{9}\left(b^{2}\right)^{3}\left(c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 2^{3}a^{9}b^{6}\left(c^{5}\right)^{3}}$$

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

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 2^{3}a^{9}b^{6}c^{15}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{27a^{5}b^{7}c^{1}}{8a^{6}b^{3}\times 8a^{9}b^{6}c^{15}}$$

Multiply 8 and 8 to get 64.

$$\frac{27a^{5}b^{7}c^{1}}{64a^{6}b^{3}a^{9}b^{6}c^{15}}$$

To multiply powers of the same base, add their exponents. Add 6 and 9 to get 15.

$$\frac{27a^{5}b^{7}c^{1}}{64a^{15}b^{3}b^{6}c^{15}}$$

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\frac{27a^{5}b^{7}c^{1}}{64a^{15}b^{9}c^{15}}$$

Cancel out ca^{5}b^{7} in both numerator and denominator.

$$\frac{27}{64b^{2}a^{10}c^{14}}$$