Question

Simplify
(1) (2a + b)³ − (2a − b)³
(2) (3r − 2k)³ + (3r + 2k)³
(3) (4a − 3)³ − (4a +3)³
(4) (5x − 7y)³ + (5x + 7y)³

Answer 1

It is known that,
${\left(a+b\right)}^3=a^3+b^3+3a^{2}b+3a{b}^2;{\left(a-b\right)}^3=a^3-b^3-3a^{2}b+3a{b}^2$
$$\begin{gathered} \left(1\right){\left(2a+b\right)}^3-{\left(2a-b\right)}^3 \\ ={\left(2a\right)}^3+{\left(b\right)}^3+3\times {\left(2a\right)}^2\times b+3\times 2a\times {\left(b\right)}^2-\left\{{\left(2a\right)}^3-{\left(b\right)}^3-3\times {\left(2a\right)}^2\times b+3\times 2a\times {\left(b\right)}^2\right\} \\ =8a^3+b^3+12a^{2}b+6a{b}^2-8a^3+b^3+12a^{2}b-6a{b}^2 \\ =2b^3+24a^{2}b \end{gathered}$$
$$\begin{gathered} \left(2\right){\left(3r-2k\right)}^3+{\left(3r+2k\right)}^3 \\ ={\left(3r\right)}^3-{\left(2k\right)}^3-3\times {\left(3r\right)}^2\times 2k+3\times 3r\times {\left(2k\right)}^2+{\left(3r\right)}^3+{\left(2k\right)}^3+3\times {\left(3r\right)}^2\times 2k+3\times 3r\times {\left(2k\right)}^2 \\ =27r^3-8k^3-54r^{2}k+36r{k}^2+27r^3+8k^3+54r^{2}k+36r{k}^2 \\ =54r^3+72r{k}^2 \end{gathered}$$
$$\begin{gathered} \left(3\right){\left(4a-3\right)}^3-{\left(4a+3\right)}^3 \\ ={\left(4a\right)}^3-{\left(3\right)}^3-3\times {\left(49\right)}^2\times 3+3\times 49\times {\left(3\right)}^2-\left\{{\left(49\right)}^3+{\left(3\right)}^3+3\times {\left(49\right)}^2\times 3+3\times 49\times {\left(3\right)}^2\right\} \\ =64a^3-27-144a^2+108-64a^3-27-144a^2-108 \\ =-54-288a^2 \end{gathered}$$
$$\begin{gathered} \left(4\right){\left(5x-7y\right)}^3+{\left(5x+7y\right)}^3 \\ ={\left(5x\right)}^3-{\left(7y\right)}^3-3\times {\left(5x\right)}^2\times 7y+3\times 5x\times {\left(7y\right)}^2+{\left(5x\right)}^3+{\left(7y\right)}^3+3\times {\left(5x\right)}^2\times \left(7y\right)+3\times 5x\times {\left(7y\right)}^2 \\ =125x^3-343y^3-525x^{2}y+735x{y}^2+125x^3+343y^3+525x^{2}y+735x{y}^2 \\ =250x^3+1470x{y}^2 \end{gathered}$$