Question

simplify them.

a)(√5 +√7) (√5-√7)

b)4√28 + 3√7

c)(√3 x√8 x 5√3) divided by√24 x√26

d)(2√7 x√8 x 5√3) divided by (3√3 x√15 x√42

e) (√2 +√3) x (√2 -√3)

Answer 1

$$\begin{gathered} \left(\mathbf{a}\right)\mathbf{.} \\ \left(\sqrt{5}+\sqrt{7}\right)\left(\sqrt{5}-\sqrt{7}\right) \\ ={\left(\sqrt{5}\right)}^2-{\left(\sqrt{7}\right)}^2\mathbf{}\left[\mathbf{using}\mathbf{}\left(\mathit{a}\mathbf{}\mathbf{+}\mathbf{}\mathit{b}\right)\mathbf{}\left(\mathit{a}\mathbf{}\mathbf{-}\mathbf{}\mathit{b}\right)\mathbf{}\mathbf{=}\mathbf{}{\mathit{a}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}{\mathit{b}}^{\mathbf{2}}\right] \\ =5-7 \\ =-2 \\ \left(\mathbf{b}\right)\mathbf{.} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}4\sqrt{28}+3\sqrt{7} \\ =4\sqrt{4\times 7}+3\sqrt{7} \\ =4\times 2\sqrt{7}+3\sqrt{7} \\ =8\sqrt{7}+3\sqrt{7} \\ =11\sqrt{7} \\ \left(\mathbf{c}\right)\mathbf{.} \\ \frac{\sqrt{3}\times \sqrt{8}\times 5\sqrt{3}}{\sqrt{24}\times \sqrt{26}} \\ =\frac{\sqrt{3}\times \sqrt{2\times 4}\times 5\sqrt{3}}{\sqrt{4\times 6}\times \sqrt{26}} \\ =\frac{\sqrt{3}\times 2\sqrt{2}\times 5\sqrt{3}}{2\sqrt{6}\times \sqrt{26}} \\ =\frac{10\sqrt{3\times 2\times 3}}{2\sqrt{6\times 26}} \\ =\frac{10\times 3\sqrt{2}}{2\sqrt{3\times 2\times 13\times 2}} \\ =\frac{30\sqrt{2}}{2\times 2\sqrt{39}} \\ =\frac{15\sqrt{2}}{2\sqrt{39}} \end{gathered}$$

$$\begin{gathered} \left(\mathbf{d}\right)\mathbf{.} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\frac{2\sqrt{7}\times \sqrt{8}\times 5\sqrt{3}}{3\sqrt{3}\times \sqrt{15}\times \sqrt{42}} \\ =\frac{2\sqrt{7}\times \sqrt{2\times 4}\times 5\sqrt{3}}{3\sqrt{3}\times \sqrt{3\times 5}\times \sqrt{2\times 3\times 7}} \\ =\frac{2\sqrt{7}\times 2\sqrt{2}\times 5\sqrt{3}}{3\sqrt{3\times 3\times 5\times 2\times 3\times 7}} \\ =\frac{20\sqrt{7\times 2\times 3}}{3\sqrt{3\times 3\times 5\times 2\times 3\times 7}} \\ =\frac{20}{3\times 3}\sqrt{\frac{7\times 2\times 3}{5\times 2\times 3\times 7}} \\ =\frac{20}{9}\sqrt{\frac{1}{5}} \\ =\frac{20}{9\sqrt{5}} \\ \left(\mathbf{e}\right)\mathbf{.} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(\sqrt{2}+\sqrt{3}\right)\times \left(\sqrt{2}-\sqrt{3}\right) \\ ={\left(\sqrt{2}\right)}^2-{\left(\sqrt{3}\right)}^2\mathbf{}\mathbf{}\left[\mathbf{as}\mathbf{}\left(\mathit{a}\mathbf{}\mathbf{+}\mathbf{}\mathit{b}\right)\mathbf{}\left(\mathit{a}\mathbf{}\mathbf{-}\mathbf{}\mathit{b}\right)\mathbf{}\mathbf{=}\mathbf{}{\mathit{a}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}{\mathit{b}}^{\mathbf{2}}\right] \\ =2-3 \\ =-1 \end{gathered}$$