Surds And Indices - Study Mode

[#81] $$left({frac{{2+sqrt 3}}{{2-sqrt3}}+ frac{{2 - sqrt 3}}{{2 + sqrt 3}} + frac{{sqrt 3 - 1}}{{sqrt 3 + 1}}}
ight)$$ xa0 xa0 xa0Simplifies to :
Correct Answer

(C) 16 - $$sqrt 3 $$

Explanation

Solution: $$left( {frac{{2 + sqrt 3 }}{{2 - sqrt 3 }} + frac{{2 - sqrt 3 }}{{2 + sqrt 3 }} + frac{{sqrt 3 - 1}}{{sqrt 3 + 1}}}
ight)$$ $$ = left{ {frac{{{{left( {2 + sqrt 3 }
ight)}^2} + {{left( {2 - sqrt 3 }
ight)}^2}}}{{left( {2 - sqrt 3 }
ight)left( {2 + sqrt 3 }
ight)}} + frac{{sqrt 3 - 1}}{{sqrt 3 + 1}} imes frac{{sqrt 3 - 1}}{{sqrt 3 - 1}}}
ight}$$ $$ = left{ {frac{{4 + 3 + 4sqrt 3 + 4 + 3 - 4sqrt 3 }}{{4 - 3}} + frac{{{{left( {sqrt 3 - 1}
ight)}^2}}}{{3 - 1}}}
ight}$$ $$eqalign{
& = left{ {14 + frac{{3 + 1 - 2sqrt 3 }}{2}}
ight} cr
& = 14 + frac{{2left( {2 - sqrt 3 }
ight)}}{2} cr
& = 14 + 2 - sqrt 3 cr
& = 16 - sqrt 3 cr} $$

[#82] The value of $$sqrt {frac{{left( {sqrt {12} - sqrt 8 }
ight)left( {sqrt 3 + sqrt 2 }
ight)}}{{5 + sqrt {24} }}} $$ xa0 xa0 xa0 is = ?
Correct Answer

(C) $$sqrt 6 - 2$$

Explanation

Solution: $$eqalign{
& sqrt {frac{{left( {sqrt {12} - sqrt 8 }
ight)left( {sqrt 3 + sqrt 2 }
ight)}}{{5 + sqrt {24} }}} cr
& = sqrt {frac{{sqrt {36} + sqrt {24} - sqrt {24} - sqrt {16} }}{{5 + sqrt {24} }}} cr
& = sqrt {frac{{6 - 4}}{{5 + sqrt {24} }}} cr
& = sqrt {frac{2}{{5 + sqrt {24} }} imes frac{{5 - sqrt {24} }}{{5 - sqrt {24} }}} cr
& = sqrt {frac{{2left( {5 - sqrt {24} }
ight)}}{{25 - 24 }}} cr
& = sqrt {2left( {5 - 2sqrt 6 }
ight)} cr
& = sqrt {2left{ {{{left( {sqrt 3 }
ight)}^2} + {{left( {sqrt 2 }
ight)}^2} - 2sqrt 3 imes sqrt 2 }
ight}} cr
& = sqrt {2{{left( {sqrt 3 - sqrt 2 }
ight)}^2}} cr
& = sqrt 2 left( {sqrt 3 - sqrt 2 }
ight) cr
& = sqrt 6 - 2 cr} $$

[#83] (19) 12 × (19) 8 ÷ (19) 4 = (19) ?
Correct Answer

6

Explanation

Solution: $$eqalign{
& frac{{{{left( {19}
ight)}^{12}} imes {{left( {19}
ight)}^8}}}{{{{left( {19}
ight)}^4}}} cr
& = frac{{{{19}^{left( {12 + 8}
ight)}}}}{{{{left( {19}
ight)}^4}}} cr
& = frac{{{{left( {19}
ight)}^{20}}}}{{{{left( {19}
ight)}^4}}} cr
& = {left( {19}
ight)^{left( {20 - 4}
ight)}} cr
& = {left( {19}
ight)^{16}} cr} $$ Hence the missing number = 16

[#84] (64) 4 ÷ (8) 5 = ?
Correct Answer

(8) 8

Explanation

Solution: $$eqalign{
& {left( {64}
ight)^4} div {left( 8
ight)^5} cr
& = {left( {{8^2}}
ight)^4} div {left( 8
ight)^5} cr
& = {left( 8
ight)^{left( {2 imes 4}
ight)}} div {8^5} cr
& = frac{{{8^8}}}{{{8^5}}} cr
& = {8^{left( {8 - 5}
ight)}} cr
& = {8^3} cr} $$

[#85] The value of $$frac{1}{{sqrt {left( {12 - sqrt {140} }
ight)} }}$$ xa0 $$ -, frac{1}{{sqrt {left( {8 - sqrt {60} }
ight)} }}$$ xa0 $$ -, frac{2}{{sqrt {left( {10 + sqrt {84} }
ight)} }}$$ xa0xa0 = is ?
Correct Answer

(A) 0

Explanation

Solution: $$frac{1}{{sqrt {left( {12 - sqrt {140} }
ight)} }} - frac{1}{{sqrt {left( {8 - sqrt {60} }
ight)} }} - frac{2}{{sqrt {left( {10 + sqrt {84} }
ight)} }}$$ $$ = frac{1}{{sqrt {left( {12 - sqrt {4 imes 35} }
ight)} }} - frac{1}{{sqrt {left( {8 - sqrt {4 imes 15} }
ight)} }} - frac{2}{{sqrt {left( {10 + sqrt {4 imes 21} }
ight)} }}$$ $$ = frac{1}{{sqrt {left( {12 - 2sqrt {35} }
ight)} }} - frac{1}{{sqrt {left( {8 - 2sqrt {15} }
ight)} }} - frac{2}{{sqrt {left( {10 + 2sqrt {21} }
ight)} }}$$ $$ = frac{1}{{sqrt {{{left( {sqrt 7 }
ight)}^2} + {{left( {sqrt 5 }
ight)}^2} - 2.sqrt 7 .sqrt 5 } }} - frac{1}{{sqrt {{{left( {sqrt 5 }
ight)}^2} + {{left( {sqrt 3 }
ight)}^2} - 2.sqrt 5 .sqrt 3 } }} - frac{2}{{sqrt {{{left( {sqrt 7 }
ight)}^2} + {{left( {sqrt 3 }
ight)}^2} + 2.sqrt 7 .sqrt 3 } }}$$ $$ = frac{1}{{sqrt {{{left( {sqrt 7 - sqrt 5 }
ight)}^2}} }} - frac{1}{{sqrt {{{left( {sqrt 5 - sqrt 3 }
ight)}^2}} }} - frac{2}{{sqrt {{{left( {sqrt 7 + sqrt 3 }
ight)}^2}} }}$$ $$ = frac{1}{{sqrt 7 - sqrt 5 }} - frac{1}{{sqrt 5 - sqrt 3 }} - frac{2}{{sqrt 7 + sqrt 3 }}$$ Rationalizing in above equation, $$ = frac{1}{{sqrt 7 - sqrt 5 }} imes frac{{sqrt 7 + sqrt 5 }}{{sqrt 7 + sqrt 5 }} - frac{1}{{sqrt 5 - sqrt 3 }} imes frac{{sqrt 5 + sqrt 3 }}{{sqrt 5 + sqrt 3 }} - frac{2}{{sqrt 7 + sqrt 3 }} imes frac{{sqrt 7 - sqrt 3 }}{{sqrt 7 - sqrt 3 }}$$ $$ = frac{{sqrt 7 + sqrt 5 }}{2} - frac{{sqrt 5 + sqrt 3 }}{2} - frac{{sqrt 7 - sqrt 3 }}{2}$$ $$ = frac{{sqrt 7 + sqrt 5 - sqrt 5 - sqrt 3 - sqrt 7 + sqrt 3 }}{2}$$ $$ = 0$$