Solve frac{(3a^{3}b)^{4}}{(2a^{2})^2}*(2ab^4)^3/9ab^3

$$\frac { ( 3 a ^ { 3 } b ) ^ { 4 } } { ( 2 a ^ { 2 } ) ^ { 2 } } \times \frac { ( 2 a b ^ { 4 } ) ^ { 3 } } { 9 a b ^ { 3 } }$$

Evaluate

$$18a^{10}b^{13}$$

Solution Steps

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

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

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

$$\frac{3^{4}a^{12}b^{4}}{\left(2a^{2}\right)^{2}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

Calculate 3 to the power of 4 and get 81.

$$\frac{81a^{12}b^{4}}{\left(2a^{2}\right)^{2}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81a^{12}b^{4}}{2^{2}\left(a^{2}\right)^{2}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81a^{12}b^{4}}{2^{2}a^{4}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{81a^{12}b^{4}}{4a^{4}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81b^{4}a^{8}}{4}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81b^{4}a^{8}}{4}\times \frac{2^{3}a^{3}\left(b^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81b^{4}a^{8}}{4}\times \frac{2^{3}a^{3}b^{12}}{9ab^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{81b^{4}a^{8}}{4}\times \frac{8a^{3}b^{12}}{9ab^{3}}$$

Cancel out ab^{3} in both numerator and denominator.

$$\frac{81b^{4}a^{8}}{4}\times \frac{8a^{2}b^{9}}{9}$$

Multiply \frac{81b^{4}a^{8}}{4} times \frac{8a^{2}b^{9}}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{81b^{4}a^{8}\times 8a^{2}b^{9}}{4\times 9}$$

Cancel out 4\times 9 in both numerator and denominator.

$$2\times 9a^{2}b^{4}a^{8}b^{9}$$

To multiply powers of the same base, add their exponents. Add 2 and 8 to get 10.

$$2\times 9a^{10}b^{4}b^{9}$$

To multiply powers of the same base, add their exponents. Add 4 and 9 to get 13.

$$2\times 9a^{10}b^{13}$$

Multiply 2 and 9 to get 18.

$$18a^{10}b^{13}$$

Expand

$$18a^{10}b^{13}$$

Solution Steps

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

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

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

$$\frac{3^{4}a^{12}b^{4}}{\left(2a^{2}\right)^{2}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

Calculate 3 to the power of 4 and get 81.

$$\frac{81a^{12}b^{4}}{\left(2a^{2}\right)^{2}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81a^{12}b^{4}}{2^{2}\left(a^{2}\right)^{2}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81a^{12}b^{4}}{2^{2}a^{4}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{81a^{12}b^{4}}{4a^{4}}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81b^{4}a^{8}}{4}\times \frac{\left(2ab^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81b^{4}a^{8}}{4}\times \frac{2^{3}a^{3}\left(b^{4}\right)^{3}}{9ab^{3}}$$

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

$$\frac{81b^{4}a^{8}}{4}\times \frac{2^{3}a^{3}b^{12}}{9ab^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{81b^{4}a^{8}}{4}\times \frac{8a^{3}b^{12}}{9ab^{3}}$$

Cancel out ab^{3} in both numerator and denominator.

$$\frac{81b^{4}a^{8}}{4}\times \frac{8a^{2}b^{9}}{9}$$

Multiply \frac{81b^{4}a^{8}}{4} times \frac{8a^{2}b^{9}}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{81b^{4}a^{8}\times 8a^{2}b^{9}}{4\times 9}$$

Cancel out 4\times 9 in both numerator and denominator.

$$2\times 9a^{2}b^{4}a^{8}b^{9}$$

To multiply powers of the same base, add their exponents. Add 2 and 8 to get 10.

$$2\times 9a^{10}b^{4}b^{9}$$

To multiply powers of the same base, add their exponents. Add 4 and 9 to get 13.

$$2\times 9a^{10}b^{13}$$

Multiply 2 and 9 to get 18.

$$18a^{10}b^{13}$$