Resolver (3+2sqrt{2})^2(3-sqrt{2})-(3+2sqrt{2})(3-2sqrt{2})^2

$$( 3 + 2 \sqrt { 2 } ) ^ { 2 } ( 3 - \sqrt { 2 } ) - ( 3 + 2 \sqrt { 2 } ) ( 3 - 2 \sqrt { 2 } ) ^ { 2 }$$

Avaliar

$$21\sqrt{2}+24\approx 53,69848481$$

Passos Para a Resolução

Utilize o teorema binomial \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para expandir \left(3+2\sqrt{2}\right)^{2}.

$$\left(9+12\sqrt{2}+4\left(\sqrt{2}\right)^{2}\right)\left(3-\sqrt{2}\right)-\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)^{2}$$

O quadrado de \sqrt{2} é 2.

$$\left(9+12\sqrt{2}+4\times 2\right)\left(3-\sqrt{2}\right)-\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)^{2}$$

Multiplique 4 e 2 para obter 8.

$$\left(9+12\sqrt{2}+8\right)\left(3-\sqrt{2}\right)-\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)^{2}$$

Some 9 e 8 para obter 17.

$$\left(17+12\sqrt{2}\right)\left(3-\sqrt{2}\right)-\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)^{2}$$

Utilize a propriedade distributiva para multiplicar 17+12\sqrt{2} por 3-\sqrt{2} e combinar termos semelhantes.

$$51+19\sqrt{2}-12\left(\sqrt{2}\right)^{2}-\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)^{2}$$

O quadrado de \sqrt{2} é 2.

$$51+19\sqrt{2}-12\times 2-\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)^{2}$$

Multiplique -12 e 2 para obter -24.

$$51+19\sqrt{2}-24-\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)^{2}$$

Subtraia 24 de 51 para obter 27.

$$27+19\sqrt{2}-\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)^{2}$$

Utilize o teorema binomial \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para expandir \left(3-2\sqrt{2}\right)^{2}.

$$27+19\sqrt{2}-\left(3+2\sqrt{2}\right)\left(9-12\sqrt{2}+4\left(\sqrt{2}\right)^{2}\right)$$

O quadrado de \sqrt{2} é 2.

$$27+19\sqrt{2}-\left(3+2\sqrt{2}\right)\left(9-12\sqrt{2}+4\times 2\right)$$

Multiplique 4 e 2 para obter 8.

$$27+19\sqrt{2}-\left(3+2\sqrt{2}\right)\left(9-12\sqrt{2}+8\right)$$

Some 9 e 8 para obter 17.

$$27+19\sqrt{2}-\left(3+2\sqrt{2}\right)\left(17-12\sqrt{2}\right)$$

Utilize a propriedade distributiva para multiplicar 3+2\sqrt{2} por 17-12\sqrt{2} e combinar termos semelhantes.

$$27+19\sqrt{2}-\left(51-2\sqrt{2}-24\left(\sqrt{2}\right)^{2}\right)$$

O quadrado de \sqrt{2} é 2.

$$27+19\sqrt{2}-\left(51-2\sqrt{2}-24\times 2\right)$$

Multiplique -24 e 2 para obter -48.

$$27+19\sqrt{2}-\left(51-2\sqrt{2}-48\right)$$

Subtraia 48 de 51 para obter 3.

$$27+19\sqrt{2}-\left(3-2\sqrt{2}\right)$$

Para calcular o oposto de 3-2\sqrt{2}, calcule o oposto de cada termo.

$$27+19\sqrt{2}-3+2\sqrt{2}$$

Subtraia 3 de 27 para obter 24.

$$24+19\sqrt{2}+2\sqrt{2}$$

Combine 19\sqrt{2} e 2\sqrt{2} para obter 21\sqrt{2}.

$$24+21\sqrt{2}$$

Fatorizar

$$3 {(7 \sqrt{2} + 8)} = 53,69848481$$