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प्रश्न
Prove that:
`(2^(1/2)xx3^(1/3)xx4^(1/4))/(10^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(4^(-3/5)xx6)=10`
उत्तर
We have to prove that `(2^(1/2)xx3^(1/3)xx4^(1/4))/(10^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(4^(-3/5)xx6)=10`
Let x = `(2^(1/2)xx3^(1/3)xx4^(1/4))/(10^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(4^(-3/5)xx6)`
`=(2^(1/2)xx3^(1/3)xx2^(2xx1/4))/(5^(-1/5)xx2^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(2^(2xx(-3)/5)xx3xx2)`
`=(2^(1/2)xx3^(1/3)xx2^(1/2))/(5^(-1/5)xx2^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(2^(2xx(-3)/5)xx3xx2)`
`=(2^(1/2+1/2+1/5)xx3^(1/3))/(5^(-1/5+3/5))div(3^(4/3-1)xx5^(-7/5))/(2^(-6/5+1))`
`=(2^((1xx5)/(2xx5)+(1xx5)/(2xx5)+(1xx2)/(2xx5))xx3^(1/3))/5^((-1+3)/5)div(3^(4/3-(1xx3)/(1xx3))xx5^(-7/5))/2^(-6/5+(1xx5)/(1xx5))`
`=(2^(5/10+5/10+2/10)xx3^(1/3))/5^(2/5)div(3^((4-3)/3)xx5^(-7/5))/2^((-6+5)/5)`
`=(2^(12/10)xx3^(1/3))/5^(2/5)div(3^(1/3)xx5^(-7/5))/2^(-1/5)`
`=(2^(12/10)xx3^(1/3))/(5^(2/5)/1)div(3^(1/3)xx5^(-7/5))/(1/2^(1/5))`
`=(2^(12/10)xx3^(1/3)xx1/5^(2/5))/(3^(1/3)/1xx1/(5^(7/5))xx2^(1/5)/1)`
`=2^(12/10)xx3^(1/3)xx1/5^(2/5)xx1/3^(1/3)xx5^(7/5)/1xx1/2^(1/5)`
`=2^(12/10)xx1/2^(1/5)xx3^(1/3)xx1/3^(1/3)xx1/5^(2/5)xx5^(7/5)/1`
`=2^(12/10)/2^(1/5)xx5^(7/5)/5^(2/5)`
`=2^(12/10-1/5)xx5^(7/5-2/5)`
`=2^(12/10-(1xx2)/(5xx2))xx5^((7-2)/5)`
`=2^((12-2)/10)xx5^(5/5)`
`=2^(10/10)xx5^(5/5)`
= 2 x 5
= 10
Hence, `(2^(1/2)xx3^(1/3)xx4^(1/4))/(10^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(4^(-3/5)xx6)=10`
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