Question

Expand
(1) ${\left(k+4\right)}^3$
(2) ${\left(7x+8y\right)}^3$
(3) ${\left(7+m\right)}^3$
(4) ${\left(52\right)}^3$
(5) ${\left(101\right)}^3$
(6) ${\left(x+\frac{1}{x}\right)}^3$
(7) ${\left(2m+\frac{1}{5}\right)}^3$
(8) ${\left(\frac{5x}{y}+\frac{y}{5x}\right)}^3$

Answer 1

It is known that,
${\left(a+b\right)}^3=a^3+b^3+3a^{2}b+3a{b}^2$
$$\begin{gathered} \left(1\right){\left(k+4\right)}^3 \\ ={\left(k\right)}^3+{\left(4\right)}^3+3\times {\left(k\right)}^2\times \left(4\right)+3\times \left(k\right)\times {\left(4\right)}^2 \\ =k^3+64+12k^2+48k \end{gathered}$$
$$\begin{gathered} \left(2\right){\left(7x+8y\right)}^3 \\ ={\left(7x\right)}^3+{\left(8y\right)}^3+3\times {\left(7x\right)}^2\times \left(8y\right)+3\times \left(7x\right)\times {\left(8y\right)}^2 \\ =343x^3+512y^3+1176x^{2}y+1344x{y}^2 \end{gathered}$$
$$\begin{gathered} \left(3\right){\left(7+m\right)}^3 \\ ={\left(7\right)}^3+{\left(m\right)}^3+3\times {\left(7\right)}^2\times \left(m\right)+3\times \left(7\right)\times {\left(m\right)}^2 \\ =343+m^3+147m+21m^2 \end{gathered}$$
$$\begin{gathered} \left(4\right){\left(52\right)}^3 \\ ={\left(50+2\right)}^3 \\ ={\left(50\right)}^3+{\left(2\right)}^3+3\times {\left(50\right)}^2\times \left(2\right)+3\times \left(50\right)\times {\left(2\right)}^2 \\ =125000+8+15000+600 \\ =140608 \end{gathered}$$
$$\begin{gathered} \left(5\right){\left(101\right)}^3 \\ ={\left(100+1\right)}^3 \\ ={\left(100\right)}^3+{\left(1\right)}^3+3\times {\left(100\right)}^2\times 1+3\times 100\times {\left(1\right)}^2 \\ =1000000+1+30000+300 \\ =1030301 \end{gathered}$$
$$\begin{gathered} \left(6\right){\left(x+\frac{1}{x}\right)}^3 \\ ={\left(x\right)}^3+{\left(\frac{1}{x}\right)}^3+3\times {\left(x\right)}^2\times \left(\frac{1}{x}\right)+3\times \left(x\right)\times {\left(\frac{1}{x}\right)}^2 \\ =x^3+\frac{1}{x^3}+3x+\frac{3}{x} \end{gathered}$$
$$\begin{gathered} \left(7\right){\left(2m+\frac{1}{5}\right)}^3 \\ ={\left(2m\right)}^3+{\left(\frac{1}{5}\right)}^3+3\times {\left(2m\right)}^2\times \left(\frac{1}{5}\right)+3\times \left(2m\right)\times {\left(\frac{1}{5}\right)}^2 \\ =8m^3+\frac{1}{125}+\frac{12m^2}{5}+\frac{6m}{25} \end{gathered}$$
$$\begin{gathered} \left(8\right){\left(\frac{5x}{y}+\frac{y}{5x}\right)}^3 \\ ={\left(\frac{5x}{y}\right)}^3+{\left(\frac{y}{5x}\right)}^3+3\times {\left(\frac{5x}{y}\right)}^2\times \left(\frac{y}{5x}\right)+3\times \left(\frac{5x}{y}\right)\times {\left(\frac{y}{5x}\right)}^2 \\ =\frac{125x^3}{y^3}+\frac{y^3}{125x^3}+\frac{15x}{y}+\frac{3y}{5x} \end{gathered}$$