Solve (5x^4+1)^2+(4x^5+4)^3

$$( 5 x ^ { 4 } + 1 ) ^ { 2 } + ( 4 x ^ { 5 } + 4 ) ^ { 3 }$$

Evaluate

$$64\left(x^{5}+1\right)^{3}+\left(5x^{4}+1\right)^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x^{4}+1\right)^{2}.

$$25\left(x^{4}\right)^{2}+10x^{4}+1+\left(4x^{5}+4\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$25x^{8}+10x^{4}+1+\left(4x^{5}+4\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(4x^{5}+4\right)^{3}.

$$25x^{8}+10x^{4}+1+64\left(x^{5}\right)^{3}+192\left(x^{5}\right)^{2}+192x^{5}+64$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$25x^{8}+10x^{4}+1+64x^{15}+192\left(x^{5}\right)^{2}+192x^{5}+64$$

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

$$25x^{8}+10x^{4}+1+64x^{15}+192x^{10}+192x^{5}+64$$

Add 1 and 64 to get 65.

$$25x^{8}+10x^{4}+65+64x^{15}+192x^{10}+192x^{5}$$

Expand

$$64x^{15}+192x^{10}+25x^{8}+192x^{5}+10x^{4}+65$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x^{4}+1\right)^{2}.

$$25\left(x^{4}\right)^{2}+10x^{4}+1+\left(4x^{5}+4\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$25x^{8}+10x^{4}+1+\left(4x^{5}+4\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(4x^{5}+4\right)^{3}.

$$25x^{8}+10x^{4}+1+64\left(x^{5}\right)^{3}+192\left(x^{5}\right)^{2}+192x^{5}+64$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$25x^{8}+10x^{4}+1+64x^{15}+192\left(x^{5}\right)^{2}+192x^{5}+64$$

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

$$25x^{8}+10x^{4}+1+64x^{15}+192x^{10}+192x^{5}+64$$

Add 1 and 64 to get 65.

$$25x^{8}+10x^{4}+65+64x^{15}+192x^{10}+192x^{5}$$