Solve 4[(-3a^{3})^{2}(-2a^{2})+10a^{8}]^2:(4a^5)^2-2a^6

$$4 [ ( - 3 a ^ { 3 } ) ^ { 2 } ( - 2 a ^ { 2 } ) + 10 a ^ { 8 } ] ^ { 2 } : ( 4 a ^ { 5 } ) ^ { 2 } - 2 a ^ { 6 }$$

Evaluate

$$14a^{6}$$

Solution Steps

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

$$\frac{4\left(\left(-3\right)^{2}\left(a^{3}\right)^{2}\left(-2\right)a^{2}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

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

$$\frac{4\left(\left(-3\right)^{2}a^{6}\left(-2\right)a^{2}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{4\left(9a^{6}\left(-2\right)a^{2}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Multiply 9 and -2 to get -18.

$$\frac{4\left(-18a^{6}a^{2}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{4\left(-18a^{8}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Combine -18a^{8} and 10a^{8} to get -8a^{8}.

$$\frac{4\left(-8a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

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

$$\frac{4\left(-8\right)^{2}\left(a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

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

$$\frac{4\left(-8\right)^{2}a^{16}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Calculate -8 to the power of 2 and get 64.

$$\frac{4\times 64a^{16}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Multiply 4 and 64 to get 256.

$$\frac{256a^{16}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Expand \left(4a^{5}\right)^{2}.

$$\frac{256a^{16}}{4^{2}\left(a^{5}\right)^{2}}-2a^{6}$$

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

$$\frac{256a^{16}}{4^{2}a^{10}}-2a^{6}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{256a^{16}}{16a^{10}}-2a^{6}$$

Cancel out 16a^{10} in both numerator and denominator.

$$16a^{6}-2a^{6}$$

Combine 16a^{6} and -2a^{6} to get 14a^{6}.

$$14a^{6}$$

Expand

$$14a^{6}$$

Solution Steps

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

$$\frac{4\left(\left(-3\right)^{2}\left(a^{3}\right)^{2}\left(-2\right)a^{2}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

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

$$\frac{4\left(\left(-3\right)^{2}a^{6}\left(-2\right)a^{2}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{4\left(9a^{6}\left(-2\right)a^{2}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Multiply 9 and -2 to get -18.

$$\frac{4\left(-18a^{6}a^{2}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{4\left(-18a^{8}+10a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Combine -18a^{8} and 10a^{8} to get -8a^{8}.

$$\frac{4\left(-8a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

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

$$\frac{4\left(-8\right)^{2}\left(a^{8}\right)^{2}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

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

$$\frac{4\left(-8\right)^{2}a^{16}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Calculate -8 to the power of 2 and get 64.

$$\frac{4\times 64a^{16}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Multiply 4 and 64 to get 256.

$$\frac{256a^{16}}{\left(4a^{5}\right)^{2}}-2a^{6}$$

Expand \left(4a^{5}\right)^{2}.

$$\frac{256a^{16}}{4^{2}\left(a^{5}\right)^{2}}-2a^{6}$$

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

$$\frac{256a^{16}}{4^{2}a^{10}}-2a^{6}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{256a^{16}}{16a^{10}}-2a^{6}$$

Cancel out 16a^{10} in both numerator and denominator.

$$16a^{6}-2a^{6}$$

Combine 16a^{6} and -2a^{6} to get 14a^{6}.

$$14a^{6}$$