Simplify the following expression
i) (√3-√2) (√3+5√2)
ii) (5+√7) (2+√5)
Answers
Step-by-step explanation:
♦ Simplification of
(i) (5 + \sqrt{7})(2 + \sqrt{5})(5+
7
)(2+
5
)
= 5( 2 + \sqrt{5} ) + \sqrt{7}(2 + \sqrt{5})=5(2+
5
)+
7
(2+
5
)
= 10 + 5 \sqrt{5} + 2 \sqrt{7} + \sqrt{ 35}=10+5
5
+2
7
+
35
(ii) (5 + \sqrt{5})(5 - \sqrt{5})(5+
5
)(5−
5
)
>> The above's equation is in the form of an Identity .
(a + b) (a - b)
• Let "5" be "a" and "\sqrt{5}
5
" be "b" .
• Then by multiplying
(a+b) (a-b)(a+b)(a−b)
= a^2 - ab + ab - b^2=a
2
−ab+ab−b
2
= a^2 - b^2=a
2
−b
2
♦ Now by substituting value
= 5^2 - \sqrt{5}^2=5
2
−
5
2
= 25 - 5=25−5
= 20=20
(iii) (\sqrt{3} + \sqrt{7} )^2(
3
+
7
)
2
♦ Now by using Identity
(a+b)^2 = a^2 + 2ab + b^2(a+b)
2
=a
2
+2ab+b
2
♦ We will take
a = \sqrt{3}
3
b = \sqrt{7}
7
♦ Then
(\sqrt{3} + \sqrt{7} )^2 = \sqrt{3}^2 + \sqrt{7}^2 + 2 \times \sqrt{3} \times \sqrt{7}(
3
+
7
)
2
=
3
2
+
7
2
+2×
3
×
7
= 3 + 7 + 2 \sqrt{21}=3+7+2
21
= 10 + 2\sqrt{21}=10+2
21
(iv) (\sqrt{11} - \sqrt{7})(\sqrt{11} + \sqrt{7})(
11
−
7
)(
11
+
7
)
>> The above's equation is in the form of an Identity .
(a - b) (a + b)
• Let "\sqrt{11}
11
" be "a" and "\sqrt{7}
7
" be "b" .
• Then by multiplying
(a - b) (a+b)(a−b)(a+b)
= a^2 + ab - ab - b^2=a
2
+ab−ab−b
2
= a^2 - b^2=a
2
−b
2
♦ Now by substituting value
= \sqrt{11}^2 - \sqrt{7}^2=
11
2
−
7
2
= 11 - 7=11−7
=4