Solve f(x)=-4(-4x^2+16)^3+5

$$f ( x ) = - 4 ( - 4 x ^ { 2 } + 16 ) ^ { 3 } + 5$$

Evaluate

$$-256\left(4-x^{2}\right)^{3}+5$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(-4x^{2}+16\right)^{3}.

$$-4\left(-64\left(x^{2}\right)^{3}+768\left(x^{2}\right)^{2}-3072x^{2}+4096\right)+5$$

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

$$-4\left(-64x^{6}+768\left(x^{2}\right)^{2}-3072x^{2}+4096\right)+5$$

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

$$-4\left(-64x^{6}+768x^{4}-3072x^{2}+4096\right)+5$$

Use the distributive property to multiply -4 by -64x^{6}+768x^{4}-3072x^{2}+4096.

$$256x^{6}-3072x^{4}+12288x^{2}-16384+5$$

Add -16384 and 5 to get -16379.

$$256x^{6}-3072x^{4}+12288x^{2}-16379$$

Expand

$$256x^{6}-3072x^{4}+12288x^{2}-16379$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(-4x^{2}+16\right)^{3}.

$$-4\left(-64\left(x^{2}\right)^{3}+768\left(x^{2}\right)^{2}-3072x^{2}+4096\right)+5$$

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

$$-4\left(-64x^{6}+768\left(x^{2}\right)^{2}-3072x^{2}+4096\right)+5$$

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

$$-4\left(-64x^{6}+768x^{4}-3072x^{2}+4096\right)+5$$

Use the distributive property to multiply -4 by -64x^{6}+768x^{4}-3072x^{2}+4096.

$$256x^{6}-3072x^{4}+12288x^{2}-16384+5$$

Add -16384 and 5 to get -16379.

$$256x^{6}-3072x^{4}+12288x^{2}-16379$$