Solve (2x^{2}+5)^{3}+18x^9(2x^2+5)^2e^5

$$( 2 x ^ { 2 } + 5 ) ^ { 3 } + 18 x ^ { 9 } ( 2 x ^ { 2 } + 5 ) ^ { 2 } e ^ { 5 }$$

Evaluate

$$\left(2x^{2}+5\right)^{2}\left(18e^{5}x^{9}+2x^{2}+5\right)$$

Solution Steps

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

$$8\left(x^{2}\right)^{3}+60\left(x^{2}\right)^{2}+150x^{2}+125+18x^{9}\left(2x^{2}+5\right)^{2}e^{5}$$

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

$$8x^{6}+60\left(x^{2}\right)^{2}+150x^{2}+125+18x^{9}\left(2x^{2}+5\right)^{2}e^{5}$$

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

$$8x^{6}+60x^{4}+150x^{2}+125+18x^{9}\left(2x^{2}+5\right)^{2}e^{5}$$

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

$$8x^{6}+60x^{4}+150x^{2}+125+18x^{9}\left(4\left(x^{2}\right)^{2}+20x^{2}+25\right)e^{5}$$

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

$$8x^{6}+60x^{4}+150x^{2}+125+18x^{9}\left(4x^{4}+20x^{2}+25\right)e^{5}$$

Use the distributive property to multiply 18x^{9} by 4x^{4}+20x^{2}+25.

$$8x^{6}+60x^{4}+150x^{2}+125+\left(72x^{13}+360x^{11}+450x^{9}\right)e^{5}$$

Use the distributive property to multiply 72x^{13}+360x^{11}+450x^{9} by e^{5}.

$$8x^{6}+60x^{4}+150x^{2}+125+72x^{13}e^{5}+360x^{11}e^{5}+450x^{9}e^{5}$$

Expand

$$72e^{5}x^{13}+360e^{5}x^{11}+450e^{5}x^{9}+8x^{6}+60x^{4}+150x^{2}+125$$

Solution Steps

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

$$8\left(x^{2}\right)^{3}+60\left(x^{2}\right)^{2}+150x^{2}+125+18x^{9}\left(2x^{2}+5\right)^{2}e^{5}$$

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

$$8x^{6}+60\left(x^{2}\right)^{2}+150x^{2}+125+18x^{9}\left(2x^{2}+5\right)^{2}e^{5}$$

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

$$8x^{6}+60x^{4}+150x^{2}+125+18x^{9}\left(2x^{2}+5\right)^{2}e^{5}$$

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

$$8x^{6}+60x^{4}+150x^{2}+125+18x^{9}\left(4\left(x^{2}\right)^{2}+20x^{2}+25\right)e^{5}$$

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

$$8x^{6}+60x^{4}+150x^{2}+125+18x^{9}\left(4x^{4}+20x^{2}+25\right)e^{5}$$

Use the distributive property to multiply 18x^{9} by 4x^{4}+20x^{2}+25.

$$8x^{6}+60x^{4}+150x^{2}+125+\left(72x^{13}+360x^{11}+450x^{9}\right)e^{5}$$

Use the distributive property to multiply 72x^{13}+360x^{11}+450x^{9} by e^{5}.

$$8x^{6}+60x^{4}+150x^{2}+125+72x^{13}e^{5}+360x^{11}e^{5}+450x^{9}e^{5}$$