Surds And Indices - Study Mode

[#116] The value of [(10) 150 ÷ (10) 146 ]
Correct Answer

(B) 10000

Explanation

Solution: $$eqalign{
& {left( {10}
ight)^{150}} div {left( {10}
ight)^{146}} = frac{{{{10}^{150}}}}{{{{10}^{146}}}} cr
& = {10^{150 - 146}} cr
& = {10^4} cr
& = 10000 cr} $$

[#117] $$frac{1}{{1 + {x^{left( {b - a}
ight)}} + {x^{left( {c - a}
ight)}}}}$$ xa0xa0 $$ + frac{1}{{1 + {x^{left( {a - b}
ight)}} + {x^{left( {c - b}
ight)}}}}$$ xa0xa0 $$ + frac{1}{{1 + {x^{left( {b - c}
ight)}} + {x^{left( {a - c}
ight)}}}} = ?$$
Correct Answer

(B) 1

Explanation

Solution: Given exp. = $$ = frac{1}{{left( {1 + frac{{{x^b}}}{{{x^a}}} + frac{{{x^c}}}{{{x^a}}}}
ight)}} + $$ xa0 $$frac{1}{{left( {1 + frac{{{x^a}}}{{{x^b}}} + frac{{{x^c}}}{{{x^b}}}}
ight)}} + $$ xa0 $$frac{1}{{left( {1 + frac{{{x^b}}}{{{x^c}}} + frac{{{x^a}}}{{{x^c}}}}
ight)}}$$ $$ = frac{{{x^a}}}{{left( {{x^a} + {x^b} + {x^c}}
ight)}} + $$ xa0 $$frac{{{x^b}}}{{left( {{x^a} + {x^b} + {x^c}}
ight)}} + $$ xa0 $$frac{{{x^c}}}{{left( {{x^a} + {x^b} + {x^c}}
ight)}}$$ $$eqalign{
& = frac{{ {{x^a} + {x^b} + {x^c}} }}{{ {{x^a} + {x^b} + {x^c}} }} cr
& = 1 cr} $$

[#118] (25) 7.5 × (5) 2.5 ÷ (125) 1.5 = 5 ?
Correct Answer

(B) 13

Explanation

Solution: $$eqalign{
& { ext{Let}},{left( {25}
ight)^{7.5}} imes {left( 5
ight)^{2.5}} div {left( {125}
ight)^{1.5}} = {5^x} cr
& { ext{Then}},,frac{{{{left( {{5^2}}
ight)}^{7.5}} imes {{left( 5
ight)}^{2.5}}}}{{{{left( {{5^3}}
ight)}^{1.5}}}} = {5^x} cr
& Rightarrow frac{{{5^{left( {2 imes 7.5}
ight)}} imes {5^{2.5}}}}{{{5^{left( {3 imes 1.5}
ight)}}}} = {5^x} cr
& Rightarrow frac{{{5^{15}} imes {5^{2.5}}}}{{{5^{4.5}}}} = {5^x} cr
& Rightarrow {5^x} = {5^{left( {15 + 2.5 - 4.5}
ight)}} cr
& Rightarrow {5^x} = {5^{13}} cr
& herefore x = 13 cr} $$

[#119] (0.04) -1.5 = ?
Correct Answer

(B) 125

Explanation

Solution: $$eqalign{
& {left( {0.04}
ight)^{ - 1.5}} = {left( {frac{4}{{100}}}
ight)^{ - 1.5}} cr
& = {left( {frac{1}{{25}}}
ight)^{ - left( {3/2}
ight)}} cr
& = {left( {25}
ight)^{left( {3/2}
ight)}} cr
& = {left( {{5^2}}
ight)^{left( {3/2}
ight)}} cr
& = {left( 5
ight)^{2 imes left( {3/2}
ight)}} cr
& = {5^3} cr
& = 125 cr} $$

[#120] Suppose 4 a = 5, 5 b = 6, 6 c = 7, 7 d = 8, then the value of abcd is = ?
Correct Answer

(B) $$frac{3}{2}$$

Explanation

Solution: $$eqalign{
& 8 = {7^d} cr
& ,,,,,, = {left( {{6^c}}
ight)^d} cr
& ,,,,,, = {left( {{5^b}}
ight)^{cd}} cr
& ,,,,,, = {5^{bcd}} cr
& ,,,,,, = {left( {{4^a}}
ight)^{bcd}} cr
& ,,,,,, = {4^{abcd}} cr
& Rightarrow {4^{abcd}} = 8 cr
& Rightarrow {left( {{2^2}}
ight)^{abcd}} = {2^3} cr
& Rightarrow 2abcd = 3 cr
& Rightarrow abcd = frac{3}{2} cr} $$