Question

Simplify each of the following expressions:
A.$(3 + \sqrt 3 )(2 + \sqrt 2 )$
B.$(3 + \sqrt 3 )(3 - \sqrt 3 )$
C.${(\sqrt 5 + \sqrt 2 )^2}$
D.$(\sqrt 5 + \sqrt 2 )(\sqrt 5 - \sqrt 2 )$

Answer 1

Hint: The algebraic identities to be used in these questions are
1.${(a + b)^2} = {a^2} + 2ab + {b^2}$
2.$(a + b)(x + y) = ax + ay + bx + by$
3.$(a + b)(a - b) = {a^2} - {b^2}$

Complete step-by-step answer:
A.The given expression is $(3 + \sqrt 3 )(2 + \sqrt 2 )$,
Since we know that $(a + b)(x + y) = ax + ay + bx + by$
Thus using this identity we get,
$(3 + \sqrt 3 )(2 + \sqrt 2 )$=$(3 \times 2) + (3 \times \sqrt 2 ) + (\sqrt 3 \times 2) + (\sqrt 3 \times \sqrt 2 )$
As $\sqrt a \times \sqrt b = \sqrt {ab} $
$ \Rightarrow $ $(3 + \sqrt 3 )(2 + \sqrt 2 )$=$6 + 3\sqrt 2 + 2\sqrt 3 + \sqrt 6 $ ………………… 1
The above expression is the required simplified expression.

B.The given expression is $(3 + \sqrt 3 )(3 - \sqrt 3 )$
Since we know that $(a + b)(a - b) = {a^2} - {b^2}$
Thus by this identity we get,
$(3 + \sqrt 3 )(3 - \sqrt 3 )$=${(3)^2} - {(\sqrt 3 )^2}$
$ \Rightarrow $ $(3 + \sqrt 3 )(3 - \sqrt 3 )$=$9 - 3$
$ \Rightarrow $$(3 + \sqrt 3 )(3 - \sqrt 3 )$=$6$ ……………………………. 2

C.The given expression is ${(\sqrt 5 + \sqrt 2 )^2}$
Since we know that ${(a + b)^2} = {a^2} + 2ab + {b^2}$
Thus by this identity we get,
${(\sqrt 5 + \sqrt 2 )^2}$=${(\sqrt 5 )^2} + 2(\sqrt 5 )(\sqrt 2 ) + {(\sqrt 2 )^2}$
$ \Rightarrow $${(\sqrt 5 + \sqrt 2 )^2}$=$5 + 2\sqrt {10} + 2$

$ \Rightarrow $${(\sqrt 5 + \sqrt 2 )^2}$=$7 + 2\sqrt {10} $ …………………………… 3

D.The given expression is $(\sqrt 5 + \sqrt 2 )(\sqrt 5 - \sqrt 2 )$
Since we know that $(a + b)(a - b) = {a^2} - {b^2}$
Thus using this identity we get,
$(\sqrt 5 + \sqrt 2 )(\sqrt 5 - \sqrt 2 )$=${(\sqrt 5 )^2} - {(\sqrt 2 )^2}$

$ \Rightarrow $$(\sqrt 5 + \sqrt 2 )(\sqrt 5 - \sqrt 2 )$=$5 - 2$
$ \Rightarrow $$(\sqrt 5 + \sqrt 2 )(\sqrt 5 - \sqrt 2 )$=$3$ ………………………. 4

Note: Some other general algebraic identities we need to remember to solve these types of questions:
1.${(a - b)^2} = {a^2} - 2ab + {b^2}$
2.$(x + a)(x + b) = {x^2} + (a + b)x + ab$
3.${(a + b)^3} = {a^3} + 3ab(a + b) + {b^3}$
4.${(a - b)^3} = {a^3} - 3ab(a - b) - {b^3}$