Square Root And Cube Root - Study Mode

[#81] Given $$sqrt 5 = 2.2361,$$ xa0 $$sqrt 3 = 1.7321{ ext{,}}$$ xa0 then $$frac{1}{{sqrt 5 - sqrt 3 }}$$ xa0 is equal to ?
Correct Answer

(C) 1.9841

Explanation

Solution: $$eqalign{
& Rightarrow frac{1}{{sqrt 5 - sqrt 3 }} cr
& = frac{1}{{sqrt 5 - sqrt 3 }} imes frac{{left( {sqrt 5 + sqrt 3 }
ight)}}{{left( {sqrt 5 + sqrt 3 }
ight)}} cr
& = frac{{left( {sqrt 5 + sqrt 3 }
ight)}}{{5 - 3}} cr
& = frac{{left( {2.2361 + 1.7321}
ight)}}{2} cr
& = frac{{3.9682}}{2} cr
& = 1.9841{ ext{ }} cr} $$

[#82] $$frac{1}{{left( {sqrt 9 - sqrt 8 }
ight)}} , - $$ xa0 $$frac{1}{{left( {sqrt 8 - sqrt 7 }
ight)}} , + $$ xa0 $$frac{1}{{left( {sqrt 7 - sqrt 6 }
ight)}} , - $$ xa0 $$frac{1}{{left( {sqrt 6 - sqrt 5 }
ight)}} , + $$ xa0 $$frac{1}{{left( {sqrt 5 - sqrt 4 }
ight)}}$$ xa0 is equal to ?
Correct Answer

(D) 5

Explanation

Solution: Given expression, $$ = frac{1}{{left( {sqrt 9 - sqrt 8 }
ight)}} imes frac{{left( {sqrt 9 + sqrt 8 }
ight)}}{{left( {sqrt 9 + sqrt 8 }
ight)}}$$ xa0 xa0 $$ - frac{1}{{left( {sqrt 8 - sqrt 7 }
ight)}}$$ xa0 $$ imes frac{{left( {sqrt 8 + sqrt 7 }
ight)}}{{left( {sqrt 8 + sqrt 7 }
ight)}}$$ xa0 $$ + frac{1}{{left( {sqrt 7 - sqrt 6 }
ight)}}$$ xa0 $$ imes frac{{left( {sqrt 7 + sqrt 6 }
ight)}}{{left( {sqrt 7 + sqrt 6 }
ight)}}$$ xa0 $$ - frac{1}{{left( {sqrt 6 - sqrt 5 }
ight)}}$$ xa0 $$ imes frac{{left( {sqrt 6 + sqrt 5 }
ight)}}{{left( {sqrt 6 + sqrt 5 }
ight)}}$$ xa0 $$ + frac{1}{{left( {sqrt 5 - sqrt 4 }
ight)}}$$ xa0 $$ imes frac{{left( {sqrt 5 + sqrt 4 }
ight)}}{{left( {sqrt 5 + sqrt 4 }
ight)}}$$ $$ = frac{{left( {sqrt 9 + sqrt 8 }
ight)}}{{left( {9 - 8}
ight)}} - frac{{left( {sqrt 8 + sqrt 7 }
ight)}}{{left( {8 - 7}
ight)}}$$ xa0 xa0 $$ + frac{{left( {sqrt 7 + sqrt 6 }
ight)}}{{left( {7 - 6}
ight)}}$$ xa0 $$ - frac{{left( {sqrt 6 + sqrt 5 }
ight)}}{{left( {6 - 5}
ight)}}$$ xa0 $$ + frac{{left( {sqrt 5 + sqrt 4 }
ight)}}{{left( {5 - 4}
ight)}}$$ $$ = left( {sqrt 9 + sqrt 8 }
ight) - left( {sqrt 8 + sqrt 7 }
ight)$$ xa0 xa0 $$ + left( {sqrt 7 + sqrt 6 }
ight)$$ xa0 $$ - left( {sqrt 6 + sqrt 5 }
ight)$$ xa0 $$ + left( {sqrt 5 + sqrt 4 }
ight)$$ $$ = left( {sqrt 9 + sqrt 4 }
ight)$$ $$ = 3 + 2$$ $$ = 5$$

[#83] Determined the value of $$frac{1}{{sqrt 1 + sqrt 2 }}{ ext{ + }}$$ xa0$$frac{1}{{sqrt 2 + sqrt 3 }}, + $$ xa0 $$frac{1}{{sqrt 3 + sqrt 4 }}, + $$ xa0 $$...... + $$ xa0 $$frac{1}{{sqrt {120} + sqrt {121} }}{ ext{ = ?}}$$
Correct Answer

(B) 10

Explanation

Solution: Given expressing, $$ = frac{1}{{sqrt 1 + sqrt 2 }}{ ext{ + }}frac{1}{{sqrt 2 + sqrt 3 }}$$ xa0 xa0 $$ + frac{1}{{sqrt 3 + sqrt 4 }}$$ xa0 $$ + ...... + $$ xa0 $$frac{1}{{sqrt {120} + sqrt {121} }}$$ $$ = frac{1}{{sqrt 2 + sqrt 1 }}{ ext{ + }}frac{1}{{sqrt 3 + sqrt 2 }}$$ xa0 xa0 $$ + frac{1}{{sqrt 4 + sqrt 3 }}$$ xa0 $$ + ...... + $$ xa0 $$frac{1}{{sqrt {121} + sqrt {120} }}$$ $$ = frac{1}{{sqrt 2 + sqrt 1 }} imes $$ xa0 $$frac{{sqrt 2 - sqrt 1 }}{{sqrt 2 - sqrt 1 }}{ ext{ + }}$$ xa0 $$frac{1}{{sqrt 3 + sqrt 2 }} imes $$ xa0 $$frac{{sqrt 3 - sqrt 2 }}{{sqrt 3 - sqrt 2 }} + $$ xa0 $$frac{1}{{sqrt 4 + sqrt 3 }} imes $$ xa0 $$frac{{sqrt 4 - sqrt 3 }}{{sqrt 4 - sqrt 3 }} + $$ xa0 $$...... + $$ xa0 $$frac{1}{{sqrt {121} + sqrt {120} }} imes $$ xa0 $$frac{{sqrt {121} - sqrt {120} }}{{sqrt {121} - sqrt {120} }}$$ $$ = frac{{sqrt 2 - sqrt 1 }}{{2 - 1}} + frac{{sqrt 3 - sqrt 2 }}{{3 - 2}}$$ xa0 xa0 $$ + frac{{sqrt 4 - sqrt 3 }}{{4 - 3}}$$ xa0 $$ + ...... + $$ xa0 $$frac{{sqrt {121} - sqrt {120} }}{{121 - 120}}$$ $$ = sqrt 2 - sqrt 1 + sqrt 3 - sqrt 2 $$ xa0 xa0 $$ + sqrt 4 - sqrt 3 $$ xa0 $$ + ...... + $$ xa0 $$sqrt {121} - sqrt {120} $$ $$ = - 1 + sqrt {121} $$ $$ = - 1 + 11$$ $$ = 10$$

[#84] If $$sqrt 2 = 1.414{ ext{,}}$$ xa0 the square root of $$frac{{sqrt 2 - 1}}{{sqrt 2 + 1}}$$ xa0 is nearest to = ?
Correct Answer

(B) 0.414

Explanation

Solution: $$eqalign{
& = frac{{sqrt 2 - 1}}{{sqrt 2 + 1}} cr
& = frac{{left( {sqrt 2 - 1}
ight)}}{{left( {sqrt 2 + 1}
ight)}} imes frac{{left( {sqrt 2 - 1}
ight)}}{{left( {sqrt 2 - 1}
ight)}} cr
& = {left( {sqrt 2 - 1}
ight)^2} cr
& herefore sqrt {frac{{sqrt 2 - 1}}{{sqrt 2 + 1}}} cr
& = left( {sqrt 2 - 1}
ight) cr
& = left( {1.414 - 1}
ight) cr
& = 0.414 cr} $$

[#85] Given that $$sqrt 3 = 1.732{ ext{,}}$$ xa0 the value of $$frac{{3 + sqrt 6 }}{{5sqrt 3 - 2sqrt {12} - sqrt {32} + sqrt {50} }}$$ xa0 xa0xa0 is ?
Correct Answer

(B) 1.732

Explanation

Solution: $$eqalign{
& { ext{Given expression,}} cr
& = frac{{3 + sqrt 6 }}{{5sqrt 3 - 2sqrt {12} - sqrt {32} + sqrt {50} }} cr
& = frac{{3 + sqrt 6 }}{{5sqrt 3 - 4sqrt 3 - 4sqrt 2 + 5sqrt 2 }} cr
& = frac{{left( {3 + sqrt 6 }
ight)}}{{left( {sqrt 3 + sqrt 2 }
ight)}} cr
& = frac{{left( {3 + sqrt 6 }
ight)}}{{left( {sqrt 3 + sqrt 2 }
ight)}} imes frac{{left( {sqrt 3 - sqrt 2 }
ight)}}{{left( {sqrt 3 - sqrt 2 }
ight)}} cr
& = frac{{3sqrt 3 - 3sqrt 2 + 3sqrt 2 - 2sqrt 3 }}{{left( {3 - 2}
ight)}} cr
& = sqrt 3 cr
& = 1.732 cr} $$