Surds And Indices - Study Mode
[#61] The value of $$frac{{{{left( {243}
ight)}^{0.13}} imes {{left( {243}
ight)}^{0.07}}}}{{{{left( 7
ight)}^{0.25}} imes {{left( {49}
ight)}^{0.075}} imes {{left( {343}
ight)}^{0.2}}}}$$ xa0 xa0 xa0is = ?
Correct Answer
(A) $$frac{3}{7}$$
Explanation
Solution: $$eqalign{
& frac{{{{left( {243}
ight)}^{0.13}} imes {{left( {243}
ight)}^{0.07}}}}{{{{left( 7
ight)}^{0.25}} imes {{left( {49}
ight)}^{0.075}} imes {{left( {343}
ight)}^{0.2}}}} cr
& = frac{{{{left( {243}
ight)}^{left( {0.13 + 0.07}
ight)}}}}{{{{left( 7
ight)}^{0.25}} imes {{left( {{7^2}}
ight)}^{0.075}} imes {{left( {{7^3}}
ight)}^{0.2}}}} cr
& = frac{{{{left( {243}
ight)}^{0.2}}}}{{{{left( 7
ight)}^{0.25}} imes {{left( 7
ight)}^{left( {2 imes 0.075}
ight)}} imes {{left( 7
ight)}^{left( {3 imes 0.2}
ight)}}}} cr
& = frac{{{{left( {{3^5}}
ight)}^{0.02}}}}{{{{left( 7
ight)}^{0.25}} imes {{left( 7
ight)}^{0.15}} imes {{left( 7
ight)}^{0.6}}}} cr
& = frac{{{{left( 3
ight)}^{left( {5 imes 0.2}
ight)}}}}{{{{left( 7
ight)}^{left( {0.25 + 0.15 + 0.6}
ight)}}}} cr
& = frac{{{3^1}}}{{{7^1}}} cr
& = frac{3}{7} cr} $$
[#62] $$frac{{{2^{n + 4}} - 2 imes {2^n}}}{{2 imes {2^{left( {n + 3}
ight)}}}} + {2^{ - 3}}$$ xa0 xa0 is equal to = ?
Correct Answer
(D) 1
Explanation
Solution: $$eqalign{
& frac{{{2^{n + 4}} - 2 imes {2^n}}}{{2 imes {2^{left( {n + 3}
ight)}}}} + {2^{ - 3}} cr
& = frac{{{2^{n + 4}} - {2^{n + 1}}}}{{{2^{left( {n + 4}
ight)}}}} + frac{1}{{{2^3}}} cr
& = frac{{{2^{n + 1}}left( {{2^3} - 1}
ight)}}{{{2^{left( {n + 4}
ight)}}}} + frac{1}{{{2^3}}} cr
& = frac{{{2^{n + 1}} imes 7}}{{{2^{n + 1}} imes {2^3}}} + frac{1}{{{2^3}}} cr
& = left( {frac{7}{8} + frac{1}{8}}
ight) cr
& = frac{8}{8} cr
& = 1 cr} $$
[#63] $$frac{{256 imes 256 - 144 imes 144}}{{112}}$$ xa0 xa0 is equal to = ?
Correct Answer
(B) 400
Explanation
Solution: $$eqalign{
& frac{{256 imes 256 - 144 imes 144}}{{112}} cr
& = frac{{{{left( {256}
ight)}^2} - {{left( {144}
ight)}^2}}}{{112}} cr
& = frac{{left( {112}
ight)left( {400}
ight)}}{{112}} cr
& = 400 cr} $$
[#64] $$sqrt {3sqrt {3sqrt {3........} } } $$ xa0xa0 is equal to = ?
Correct Answer
(B) 3
Explanation
Solution: Shortcut method: When the question is in form $$eqalign{
& Rightarrow sqrt {nsqrt {nsqrt n } } ........infty cr
& Rightarrow { ext{So }}n{ ext{ is answer }} cr
& Rightarrow { ext{3}} cr} $$
[#65] The value of $$frac{{{2^{n - 1}} - {2^n}}}{{{2^{n + 4}} + {2^{n + 1}}}}{ ext{is = ?}}$$
Correct Answer
(A) $$ - frac{1}{{36}}$$
Explanation
Solution: $$eqalign{
& frac{{{2^{n - 1}} - {2^n}}}{{{2^{n + 4}} + {2^{n + 1}}}} cr
& = frac{{{2^{n - 1}}left( {1 - 2}
ight)}}{{{2^{n + 1}}left( {{2^3} + 1}
ight)}} cr
& = left( { - frac{1}{9}}
ight){.2^{left( {n - 1}
ight) - left( {n + 1}
ight)}} cr
& = left( { - frac{1}{9}}
ight){.2^{ - 2}} cr
& = left( { - frac{1}{9}}
ight).frac{1}{{{2^2}}} cr
& = left( { - frac{1}{9}}
ight) imes frac{1}{4} cr
& = - frac{1}{{36}} cr} $$