Solve [(-8/5a^{3}b^{4})(-5/4a^{2}b^{4})]^2:[(-2a)(a^3b^4)^3]

$$[ ( - \frac { 8 } { 5 } a ^ { 3 } b ^ { 4 } ) ( - \frac { 5 } { 4 } a ^ { 2 } b ^ { 4 } ) ] ^ { 2 } : [ ( - 2 a ) ( a ^ { 3 } b ^ { 4 } ) ^ { 3 } ]$$

Evaluate

$$-2b^{4}$$

Solution Steps

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

$$\frac{\left(-\frac{8}{5}a^{5}b^{4}\left(-\frac{5}{4}\right)b^{4}\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$\frac{\left(-\frac{8}{5}a^{5}b^{8}\left(-\frac{5}{4}\right)\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

Multiply -\frac{8}{5} and -\frac{5}{4} to get 2.

$$\frac{\left(2a^{5}b^{8}\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

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

$$\frac{2^{2}\left(a^{5}\right)^{2}\left(b^{8}\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

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

$$\frac{2^{2}a^{10}\left(b^{8}\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and 2 to get 16.

$$\frac{2^{2}a^{10}b^{16}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4a^{10}b^{16}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

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

$$\frac{4a^{10}b^{16}}{-2a\left(a^{3}\right)^{3}\left(b^{4}\right)^{3}}$$

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

$$\frac{4a^{10}b^{16}}{-2aa^{9}\left(b^{4}\right)^{3}}$$

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

$$\frac{4a^{10}b^{16}}{-2aa^{9}b^{12}}$$

To multiply powers of the same base, add their exponents. Add 1 and 9 to get 10.

$$\frac{4a^{10}b^{16}}{-2a^{10}b^{12}}$$

Cancel out 2a^{10}b^{12} in both numerator and denominator.

$$\frac{2b^{4}}{-1}$$

Anything divided by -1 gives its opposite.

$$-2b^{4}$$

Expand

$$-2b^{4}$$

Solution Steps

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

$$\frac{\left(-\frac{8}{5}a^{5}b^{4}\left(-\frac{5}{4}\right)b^{4}\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$\frac{\left(-\frac{8}{5}a^{5}b^{8}\left(-\frac{5}{4}\right)\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

Multiply -\frac{8}{5} and -\frac{5}{4} to get 2.

$$\frac{\left(2a^{5}b^{8}\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

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

$$\frac{2^{2}\left(a^{5}\right)^{2}\left(b^{8}\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

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

$$\frac{2^{2}a^{10}\left(b^{8}\right)^{2}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and 2 to get 16.

$$\frac{2^{2}a^{10}b^{16}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4a^{10}b^{16}}{-2a\left(a^{3}b^{4}\right)^{3}}$$

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

$$\frac{4a^{10}b^{16}}{-2a\left(a^{3}\right)^{3}\left(b^{4}\right)^{3}}$$

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

$$\frac{4a^{10}b^{16}}{-2aa^{9}\left(b^{4}\right)^{3}}$$

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

$$\frac{4a^{10}b^{16}}{-2aa^{9}b^{12}}$$

To multiply powers of the same base, add their exponents. Add 1 and 9 to get 10.

$$\frac{4a^{10}b^{16}}{-2a^{10}b^{12}}$$

Cancel out 2a^{10}b^{12} in both numerator and denominator.

$$\frac{2b^{4}}{-1}$$

Anything divided by -1 gives its opposite.

$$-2b^{4}$$