Simplification - Study Mode

[#106] What is the value of $$frac{{sqrt {24} + sqrt {216} }}{{sqrt {96} }} = ?$$
Correct Answer

(C) 2

Explanation

Solution: $$eqalign{
& { ext{According to question,}} cr
& ,,frac{{sqrt {24} + sqrt {216} }}{{sqrt {96} }} cr
& = frac{{sqrt {2 imes 2 imes 6} + sqrt {6 imes 6 imes 6} }}{{sqrt {4 imes 4 imes 6} }} cr
& = frac{{2sqrt 6 + 6sqrt 6 }}{{4sqrt 6 }} cr
& = frac{8}{4} cr
& = 2 cr} $$

[#107] Simplify : $$sqrt {3 + frac{{33}}{{64}}} div $$ xa0 $$sqrt {9 + frac{1}{7}} imes $$ xa0 $$2sqrt {3frac{1}{9}} $$ xa0 = ?
Correct Answer

(D) $$2frac{3}{{16}}$$

Explanation

Solution: $$eqalign{
& { ext{According to question,}} cr
& sqrt {3 + frac{{33}}{{64}}} div sqrt {9 + frac{1}{7}} imes 2sqrt {3frac{1}{9}} cr
& = sqrt {frac{{225}}{{64}}} div sqrt {frac{{64}}{7}} imes 2sqrt {frac{{28}}{9}} cr
& = sqrt {frac{{225}}{{64}} imes frac{7}{{64}} imes frac{{28}}{9}} imes 2 cr
& = frac{{5 imes 7}}{{8 imes 4}} imes 2 cr
& = frac{{35}}{{16}} cr
& = 2frac{3}{{16}} cr} $$

[#108] $$frac{{{{left( {3frac{2}{3}}
ight)}^2} - {{left( {2frac{1}{2}}
ight)}^2}}}{{{{left( {4frac{3}{4}}
ight)}^2} - {{left( {3frac{1}{3}}
ight)}^2}}}$$ xa0 $$ div $$ $$frac{{3frac{2}{3} - 2frac{1}{2}}}{{4frac{3}{4} - 3frac{1}{3}}}$$ xa0 = ?
Correct Answer

(B) $$frac{{74}}{{97}}$$

Explanation

Solution: $$eqalign{
& { ext{If }}a = 3frac{2}{3}, cr
& b = 2frac{1}{2}, cr
& c = 4frac{3}{4}, cr
& d = 3frac{1}{3}, cr
& { ext{then}} cr
& { ext{Given expression ,}} cr
& ,,,,frac{{left( {{a^2} - {b^2}}
ight)}}{{left( {{c^2} - {d^2}}
ight)}} div frac{{left( {a - b}
ight)}}{{left( {c - d}
ight)}} cr
& = frac{{left( {{a^2} - {b^2}}
ight)}}{{left( {{c^2} - {d^2}}
ight)}} imes frac{{left( {c - d}
ight)}}{{left( {a - b}
ight)}} cr
& = frac{{left( {a + b}
ight)}}{{left( {c + d}
ight)}} cr
& = frac{{3frac{2}{3} + 2frac{1}{2}}}{{4frac{3}{4} + 3frac{1}{3}}} cr
& = frac{{frac{{11}}{3} + frac{5}{2}}}{{frac{{19}}{4} + frac{{10}}{3}}} cr
& = frac{{37}}{6} imes frac{{12}}{{97}} cr
& = frac{{74}}{{97}} cr} $$

[#109] The simplified value of $$frac{{left( {1 + frac{1}{{1 + frac{1}{{100}}}}}
ight)left( {1 + frac{1}{{1 + frac{1}{{100}}}}}
ight) - left( {1 - frac{1}{{1 + frac{1}{{100}}}}}
ight)left( {1 - frac{1}{{1 + frac{1}{{100}}}}}
ight)}}{{left( {1 + frac{1}{{1 + frac{1}{{100}}}}}
ight) + left( {1 - frac{1}{{1 + frac{1}{{100}}}}}
ight)}} = ?$$
Correct Answer

(B) $$frac{{200}}{{101}}$$

Explanation

Solution: $$eqalign{
& { ext{Given expression ,}} cr
& frac{{{a^2} - {b^2}}}{{a + b}} = a - b cr
& = left( {1 + frac{1}{{1 + frac{1}{{100}}}}}
ight) - left( {1 - frac{1}{{1 + frac{1}{{100}}}}}
ight) cr
& = 2 imes frac{1}{{left( {101/100}
ight)}} cr
& = 2 imes frac{{100}}{{101}} cr
& = frac{{200}}{{101}} cr} $$

[#110] If $$frac{{547.527}}{{0.0082}}{ ext{ = }}x{ ext{,}}$$ xa0 then the value of $$frac{{547527}}{{82}}$$ xa0 is = ?
Correct Answer

(D) $$frac{x}{{10}}$$

Explanation

Solution: $$eqalign{
& { ext{According to question,}} cr
& ,,,,,,frac{{547.527}}{{0.0082}}{ ext{ = }}x{ ext{ }} cr
& Rightarrow frac{{547527}}{{82}} imes 10 = x cr
& Rightarrow frac{{547527}}{{82}} = frac{x}{{10}} cr} $$