General Mathematics

Question:`((2^(1/2) xx 3^(1/3) xx 4^(1/4))/(10^(- 1/5) xx 5^(3/5))// (3^(4/3) xx 5^(- 7/5))/(4^(- 3/5) xx 6))` 
Answer`(2^(1/2) xx 3^(1/3) xx 4^(1/4))/(10^(- 1/5) xx 5^(3/5)) xx (4^(- 3/5) xx 6)/(3^(4/3) xx 5^(- 7/5))`

= `(2^(1/2) xx 3^(1/3) xx 2^(1/2))/(2^(- 1/5) xx 5^(- 1/5) xx 5^(3/5)) xx ((2^2)^(- 3/5) xx (2 xx 3))/(3^(4/3) xx 5^(- 7/5))`

= `(2^(1/2) xx 3^(1/3) xx 2^(1/2))/(2^(- 1/5) xx 5^(- 1/5) xx 5^(3/5)) xx (2^(- 6/5) xx 2 xx 3)/(3^(4/3) xx 5^(- 7/5))`

= `(2^(1/2 + 1/2 +1/5 - 6/5 +1) . 3^(1/3 +1 - 4/3))/5^(-1/5 + 3/5 - 7/5)`

= `(2^1 xx 3^0 xx 5^1)`

= (2 x 1 x 5)
= 10

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Question:Which is largest: `root(3)(4)` or `root(4)(5)` 
Answer`:. \ root(3)(4)`

 = `4^(1/3)`

 = `4^((1/3 xx 3/3))`

 = `4^(4/12)`

 = `(4^4)^(1/12)`

 = `256^(1/12)`

`:. \ root(4)(5)`

 = `5^(1/4)`

 = `4^((1/4 xx 3/3))`

 = `5^(3/12)`

 = `(4^3)^(1/12)`

 = `125^(1/12)`

 Clearly. `(256)^(1/12)` > `(125)^(1/12)`

 or `root(3)(4)` > `root(4)(5)`

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Question:Express each of the following as a fraction:

(i) 56%
(i) 4%
(i) 0.6% 
Answer(i) 56% =`56/100` =`14/25`
(ii) 6% = `6/100` = 0.06
(iii) 0.6% = `0.6/100` =`3/500`

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Question:Express each of the following as a decimal:

(i) 28%
(ii) 6%
(iii) 0.2%
(iv) 0.04% 
Answer(i) 28% = `28/100`=0.28

(ii) 6% = `6/100` =0.06

(iii) 0.2% = `0.2/100` =0.002

(iv) 0.04% = `0.04/100` = 0.0004

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Question:Express each of following as rate per cent:

(i) `23/36`  (ii) .004  (iii) `6 3/4` 
Answer(i) `23/36` = `(23/36 xx 100)`% = `(575/9)`% = `63 8/9` %.

(ii) .004 = `4/1000` = `(4/1000 xx 100)`% = .4 %.

(iii) `6 3/4` = `27/4` = `(27/4 xx 100)`% = 675 %.

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Question:`((2^(1/2) xx 3^(1/3) xx 4^(1/4))/(10^(- 1/5) xx 5^(3/5))// (3^(4/3) xx 5^(- 7/5))/(4^(- 3/5) xx 6))`

Question:Which is largest: `root(3)(4)` or `root(4)(5)`

Question:Express each of the following as a fraction: (i) 56% (i) 4% (i) 0.6%

Question:Express each of the following as a decimal: (i) 28% (ii) 6% (iii) 0.2% (iv) 0.04%

Question:Express each of following as rate per cent: (i) `23/36` (ii) .004 (iii) `6 3/4`