Solve frac{(3a^{2}b^{2}c^{3})^{2}*(2ab)^3}{(4a^3bc^2)^2}

$$\frac { ( 3 a ^ { 2 } b ^ { 2 } c ^ { 3 } ) ^ { 2 } \cdot ( 2 a b ) ^ { 3 } } { ( 4 a ^ { 3 } b c ^ { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{9ac^{2}b^{5}}{2}$$

Solution Steps

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

$$\frac{3^{2}\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}\left(c^{3}\right)^{2}\times \left(2ab\right)^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

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

$$\frac{3^{2}a^{4}\left(b^{2}\right)^{2}\left(c^{3}\right)^{2}\times \left(2ab\right)^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

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

$$\frac{3^{2}a^{4}b^{4}\left(c^{3}\right)^{2}\times \left(2ab\right)^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

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

$$\frac{3^{2}a^{4}b^{4}c^{6}\times \left(2ab\right)^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 3 and get 8.

$$\frac{9a^{4}b^{4}c^{6}\times 8a^{3}b^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

Multiply 9 and 8 to get 72.

$$\frac{72a^{4}b^{4}c^{6}a^{3}b^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{72a^{7}b^{4}c^{6}b^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{72a^{7}b^{7}c^{6}}{\left(4a^{3}bc^{2}\right)^{2}}$$

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

$$\frac{72a^{7}b^{7}c^{6}}{4^{2}\left(a^{3}\right)^{2}b^{2}\left(c^{2}\right)^{2}}$$

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

$$\frac{72a^{7}b^{7}c^{6}}{4^{2}a^{6}b^{2}\left(c^{2}\right)^{2}}$$

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

$$\frac{72a^{7}b^{7}c^{6}}{4^{2}a^{6}b^{2}c^{4}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{72a^{7}b^{7}c^{6}}{16a^{6}b^{2}c^{4}}$$

Cancel out 8b^{2}c^{4}a^{6} in both numerator and denominator.

$$\frac{9ac^{2}b^{5}}{2}$$

Expand

$$\frac{9ac^{2}b^{5}}{2}$$

Solution Steps

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

$$\frac{3^{2}\left(a^{2}\right)^{2}\left(b^{2}\right)^{2}\left(c^{3}\right)^{2}\times \left(2ab\right)^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

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

$$\frac{3^{2}a^{4}\left(b^{2}\right)^{2}\left(c^{3}\right)^{2}\times \left(2ab\right)^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

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

$$\frac{3^{2}a^{4}b^{4}\left(c^{3}\right)^{2}\times \left(2ab\right)^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

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

$$\frac{3^{2}a^{4}b^{4}c^{6}\times \left(2ab\right)^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 3 and get 8.

$$\frac{9a^{4}b^{4}c^{6}\times 8a^{3}b^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

Multiply 9 and 8 to get 72.

$$\frac{72a^{4}b^{4}c^{6}a^{3}b^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{72a^{7}b^{4}c^{6}b^{3}}{\left(4a^{3}bc^{2}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{72a^{7}b^{7}c^{6}}{\left(4a^{3}bc^{2}\right)^{2}}$$

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

$$\frac{72a^{7}b^{7}c^{6}}{4^{2}\left(a^{3}\right)^{2}b^{2}\left(c^{2}\right)^{2}}$$

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

$$\frac{72a^{7}b^{7}c^{6}}{4^{2}a^{6}b^{2}\left(c^{2}\right)^{2}}$$

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

$$\frac{72a^{7}b^{7}c^{6}}{4^{2}a^{6}b^{2}c^{4}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{72a^{7}b^{7}c^{6}}{16a^{6}b^{2}c^{4}}$$

Cancel out 8b^{2}c^{4}a^{6} in both numerator and denominator.

$$\frac{9ac^{2}b^{5}}{2}$$