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Simplify each of the following expressions:\newline(i) (3+3)(2+2)(3+\sqrt{3})(2+\sqrt{2})\newline(ii) (3+3)(3−3)(3+\sqrt{3})(3-\sqrt{3})\newline(iii) (5+2)2(\sqrt{5}+\sqrt{2})^{2}\newline(iv) (5−2)(5+2)(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})

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Full solution

Q. Simplify each of the following expressions:\newline(i) (3+3)(2+2)(3+\sqrt{3})(2+\sqrt{2})\newline(ii) (3+3)(3−3)(3+\sqrt{3})(3-\sqrt{3})\newline(iii) (5+2)2(\sqrt{5}+\sqrt{2})^{2}\newline(iv) (5−2)(5+2)(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})
  1. Distributive Property Simplification: (i) Simplify (3+3)(2+2)(3+\sqrt{3})(2+\sqrt{2}) using the distributive property.\newline(3+3)(2+2)=3⋅2+3⋅2+3⋅2+3⋅2(3+\sqrt{3})(2+\sqrt{2}) = 3\cdot 2 + 3\cdot\sqrt{2} + \sqrt{3}\cdot 2 + \sqrt{3}\cdot\sqrt{2}\newline=6+32+23+6= 6 + 3\sqrt{2} + 2\sqrt{3} + \sqrt{6}
  2. Difference of Squares Simplification: (ii) Simplify (3+3)(3−3)(3+\sqrt{3})(3-\sqrt{3}) using the difference of squares formula.\newline(3+\sqrt{3})(3-\sqrt{3}) = 3^2 - (\sqrt{3})^2\(\newline= 9 - 3\newline= 6\)
  3. Binomial Expansion Simplification: (iii) Simplify (5+2)2(\sqrt{5}+\sqrt{2})^{2} using the binomial expansion.\newline(\sqrt{5}+\sqrt{2})^2 = (\sqrt{5})^2 + 2\cdot\sqrt{5}\cdot\sqrt{2} + (\sqrt{2})^2\(\newline= 5 + 2\sqrt{10} + 2\newline= 7 + 2\sqrt{10}\)
  4. Difference of Squares Simplification: (iv) Simplify (5−2)(5+2)(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2}) using the difference of squares formula.\newline(\sqrt{\(5\)}-\sqrt{\(2\)})(\sqrt{\(5\)}+\sqrt{\(2\)}) = (\sqrt{\(5\)})^\(2 - (\sqrt{22})^22\newline= 55 - 22\newline= 33
  5. Difference of Squares Simplification: (iv) Simplify (5−2)(5+2)(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2}) using the difference of squares formula.(5−2)(5+2)=(5)2−(2)2(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2=5−2= 5 - 2=3= 3The last part of the problem seems to be cut off and unrelated to the simplification tasks. We'll ignore it.

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