Solve frac{2a^{2}b^{3}}{(2a^{4}b^{2})^{3}}*frac{(2^{3}ab^2)^5}{2^7a^3b}

$$\frac { 2 a ^ { 2 } b ^ { 3 } } { ( 2 a ^ { 4 } b ^ { 2 } ) ^ { 3 } } \times \frac { ( 2 ^ { 3 } a b ^ { 2 } ) ^ { 5 } } { 2 ^ { 7 } a ^ { 3 } b }$$

Evaluate

$$\frac{64b^{6}}{a^{8}}$$

Solution Steps

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

$$\frac{2a^{2}b^{3}}{2^{3}\left(a^{4}\right)^{3}\left(b^{2}\right)^{3}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

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

$$\frac{2a^{2}b^{3}}{2^{3}a^{12}\left(b^{2}\right)^{3}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

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

$$\frac{2a^{2}b^{3}}{2^{3}a^{12}b^{6}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{2a^{2}b^{3}}{8a^{12}b^{6}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

Cancel out 2a^{2}b^{3} in both numerator and denominator.

$$\frac{1}{4b^{3}a^{10}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{1}{4b^{3}a^{10}}\times \frac{\left(8ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

Expand \left(8ab^{2}\right)^{5}.

$$\frac{1}{4b^{3}a^{10}}\times \frac{8^{5}a^{5}\left(b^{2}\right)^{5}}{2^{7}a^{3}b}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{1}{4b^{3}a^{10}}\times \frac{8^{5}a^{5}b^{10}}{2^{7}a^{3}b}$$

Calculate 8 to the power of 5 and get 32768.

$$\frac{1}{4b^{3}a^{10}}\times \frac{32768a^{5}b^{10}}{2^{7}a^{3}b}$$

Calculate 2 to the power of 7 and get 128.

$$\frac{1}{4b^{3}a^{10}}\times \frac{32768a^{5}b^{10}}{128a^{3}b}$$

Cancel out 128ba^{3} in both numerator and denominator.

$$\frac{1}{4b^{3}a^{10}}\times 256a^{2}b^{9}$$

Express \frac{1}{4b^{3}a^{10}}\times 256 as a single fraction.

$$\frac{256}{4b^{3}a^{10}}a^{2}b^{9}$$

Express \frac{256}{4b^{3}a^{10}}a^{2} as a single fraction.

$$\frac{256a^{2}}{4b^{3}a^{10}}b^{9}$$

Cancel out 4a^{2} in both numerator and denominator.

$$\frac{64}{b^{3}a^{8}}b^{9}$$

Express \frac{64}{b^{3}a^{8}}b^{9} as a single fraction.

$$\frac{64b^{9}}{b^{3}a^{8}}$$

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

$$\frac{64b^{6}}{a^{8}}$$

Expand

$$\frac{64b^{6}}{a^{8}}$$

Solution Steps

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

$$\frac{2a^{2}b^{3}}{2^{3}\left(a^{4}\right)^{3}\left(b^{2}\right)^{3}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

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

$$\frac{2a^{2}b^{3}}{2^{3}a^{12}\left(b^{2}\right)^{3}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

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

$$\frac{2a^{2}b^{3}}{2^{3}a^{12}b^{6}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{2a^{2}b^{3}}{8a^{12}b^{6}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

Cancel out 2a^{2}b^{3} in both numerator and denominator.

$$\frac{1}{4b^{3}a^{10}}\times \frac{\left(2^{3}ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{1}{4b^{3}a^{10}}\times \frac{\left(8ab^{2}\right)^{5}}{2^{7}a^{3}b}$$

Expand \left(8ab^{2}\right)^{5}.

$$\frac{1}{4b^{3}a^{10}}\times \frac{8^{5}a^{5}\left(b^{2}\right)^{5}}{2^{7}a^{3}b}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{1}{4b^{3}a^{10}}\times \frac{8^{5}a^{5}b^{10}}{2^{7}a^{3}b}$$

Calculate 8 to the power of 5 and get 32768.

$$\frac{1}{4b^{3}a^{10}}\times \frac{32768a^{5}b^{10}}{2^{7}a^{3}b}$$

Calculate 2 to the power of 7 and get 128.

$$\frac{1}{4b^{3}a^{10}}\times \frac{32768a^{5}b^{10}}{128a^{3}b}$$

Cancel out 128ba^{3} in both numerator and denominator.

$$\frac{1}{4b^{3}a^{10}}\times 256a^{2}b^{9}$$

Express \frac{1}{4b^{3}a^{10}}\times 256 as a single fraction.

$$\frac{256}{4b^{3}a^{10}}a^{2}b^{9}$$

Express \frac{256}{4b^{3}a^{10}}a^{2} as a single fraction.

$$\frac{256a^{2}}{4b^{3}a^{10}}b^{9}$$

Cancel out 4a^{2} in both numerator and denominator.

$$\frac{64}{b^{3}a^{8}}b^{9}$$

Express \frac{64}{b^{3}a^{8}}b^{9} as a single fraction.

$$\frac{64b^{9}}{b^{3}a^{8}}$$

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

$$\frac{64b^{6}}{a^{8}}$$