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

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

Evaluate

$$256b^{28}a^{32}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

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

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

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

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

$$\left(\frac{2^{3}a^{9}\left(b^{4}\right)^{3}}{2ab^{5}}\right)^{4}$$

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

$$\left(\frac{2^{3}a^{9}b^{12}}{2ab^{5}}\right)^{4}$$

Calculate 2 to the power of 3 and get 8.

$$\left(\frac{8a^{9}b^{12}}{2ab^{5}}\right)^{4}$$

Cancel out 2ab^{5} in both numerator and denominator.

$$\left(4b^{7}a^{8}\right)^{4}$$

Expand \left(4b^{7}a^{8}\right)^{4}.

$$4^{4}\left(b^{7}\right)^{4}\left(a^{8}\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 4 to get 28.

$$4^{4}b^{28}\left(a^{8}\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 8 and 4 to get 32.

$$4^{4}b^{28}a^{32}$$

Calculate 4 to the power of 4 and get 256.

$$256b^{28}a^{32}$$

Expand

$$256b^{28}a^{32}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

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

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

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

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

$$\left(\frac{2^{3}a^{9}\left(b^{4}\right)^{3}}{2ab^{5}}\right)^{4}$$

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

$$\left(\frac{2^{3}a^{9}b^{12}}{2ab^{5}}\right)^{4}$$

Calculate 2 to the power of 3 and get 8.

$$\left(\frac{8a^{9}b^{12}}{2ab^{5}}\right)^{4}$$

Cancel out 2ab^{5} in both numerator and denominator.

$$\left(4b^{7}a^{8}\right)^{4}$$

Expand \left(4b^{7}a^{8}\right)^{4}.

$$4^{4}\left(b^{7}\right)^{4}\left(a^{8}\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 4 to get 28.

$$4^{4}b^{28}\left(a^{8}\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 8 and 4 to get 32.

$$4^{4}b^{28}a^{32}$$

Calculate 4 to the power of 4 and get 256.

$$256b^{28}a^{32}$$