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Question: 

Question: 

Question:১১. প্রমাণ কর: `(4^n - 1)/(2^n - 1) = 2^n + 1` 

সমাধান:

 বামপক্ষ `= (4^n - 1)/(2^n - 1)`

          `= ((2^2)^n - 1)/(2^n - 1)`

         `= ((2^n)^2 - 1)/(2^n - 1)`

         `= ((2^n + 1)(2^n - 1))/((2^n - 1))` 
                              [ `:. a^2 - b^2 = (a + b)(a - b)` ]

         `= 2^n + 1`

           = ডানপক্ষ

    `:. (4^n - 1)/(2^n - 1) = 2^n + 1`  (প্রমাণিত)

Question:১২. প্রমাণ কর: `(2^(p + 1) .3^(2p - q) .5^(p + q) .6^q)/(6^p .10^(q + 2) .15^p) = 1/50` 

সমাধান: 
 বামপক্ষ `= (2^(p + 1) .3^(2p - q) .5^(p + q) .6^q)/(6^p .10^(q + 2) .15^p)`

       `= (2^(p + 1) .3^(2p - q) .5^(p + q) .(2 xx 3)^q)/((2 xx 3)^p .(5 xx 2)^(q + 2) .(3 xx 5)^q)`

       `= (2^(p + 1) .3^(2p - q) .5^(p + q) .2^q .3^q)/(2^p .3^p .5^(q + 2) .2^(q + 2) .3^p .5^p)`

       `= (2^(p + q + 1) .3^(2p - q - q) .5^(p + q))/(2^(p + q + 2) .3^(p + p) .5^(q + p + 2))`
                    
       `= (2^(p + q + 1) .3^(2p) .5^(p + q))/(2^(p + q + 2) .3^(2p) .5^(p + q + 2))`

       `= 2^((p + q + 1) - (p + q + 2)) .3^(2p - 2p) .5^((p + q) - (p + q + 2))`

       `= 2^(p + q + 1 - p - q - 2) .3^0 .5^(p + q - p - q - 2)`
 
       `= 2^(- 1) . 1 . 5^(- 2)`
 
       `= 1/2 . 1 . 1/5^2`
    
       `= 1/2 . 1 . 1/25`

       `= 1/50`

  `:. (2^(p + 1) .3^(2p - q) .5^(p + q) .6^q)/(6^p .10^(q + 2) .15^p) `

      `= 1/50`   প্রমাণিত

Question:১৩.প্রমাণ কর: `(a^1/a^m)^n . (a^m/a^n)^1 . (a^n/a^1)^m = 1` 

সমাধান: 
  বামপক্ষ 
    `=(a^1/a^m)^n . (a^m/a^n)^1 . (a^n/a^1)^m`

    `=a^(n1)/a^(mn) . a^(1m)/a^(1n) . a^(mn)/a^(m1)`

    `= a^(n1 - nm +1m - n1 + nm - 1m)`

    `= a^0`

    `= 1`

    `=`ডানপক্ষ

  `:. (a^1/a^m)^n . (a^m/a^n)^1 . (a^n/a^1)^m `

    `= 1`  (প্রমাণিত)বিকল্প সমাধান: বামপক্ষ 
  `=(a^1/a^m)^n . (a^m/a^n)^1 . (a^n/a^1)^m`

  `=(a^(1 - m))^n . (a^(m - n))^1 . (a^(n - 1))^m`

  `= (a^(n1 - mn)) . (a^(m1 - 1n)) . (a^(mn - m1))`

  `=  a^(n1 - nm + 1m - n1 + nm - 1m)`

  `= a^0`

  `= 1`

  `=`ডানপক্ষ

  `:. (a^1/a^m)^n . (a^m/a^n)^1 . (a^n/a^1)^m `

    `= 1` (প্রমাণিত)

Question:১৪.প্রমাণ কর: `a^(p + q)/a^(2r) xx a^(q + r)/a^(2p) xx a^(r + p)/a^(2q) = 1` 

সমাধান: 
  বামপক্ষ 
    `= a^(p + q)/a^(2r) xx a^(q + r)/a^(2p) xx a^(r + p)/a^(2q)`

    `= a^(p + q - 2r) . a^(q + r - 2p) . a^(r + p - 2q)`

    `= a^(p + q - 2r + q + r - 2p + r + p - 2q)`

    `= a^(2p - 2p + 2q - 2q + 2r - 2r)`

    `= a^0`

    `= 1`

    `=`ডানপক্ষ

   `:. a^(p + q)/a^(2r) xx a^(q + r)/a^(2p) xx a^(r + p)/a^(2q) `

   `= 1` (প্রমাণিত)

Question:১৫.প্রমাণ কর: `(x^a/x^b)^(1/(ab)) . (x^b/x^c)^(1/(bc)) . (x^c/x^a)^(1/(ca)) = 1` 

সমাধান:
  বামপক্ষ 
   `= (x^a/x^b)^(1/(ab)) . (x^b/x^c)^(1/(bc)) . (x^c/x^a)^(1/(ca)) `

   `= (x^(a - b))^(1/(ab)) . (x^(b - c))^(1/(bc)) . (x^(c - a))^(1/(ca))`

   `= x^((a - b)/(ab)) . x^((b - c)/(bc)) . x^((c - a)/(ca))`

   `= x^((a - b)/(ab) + (b - c)/(bc) + (c - a)/(ca))`

   `= x^((ca - bc + ab - ca + bc - ab)/(abc))`

   `= x^(0/(abc))`

   `= x^0`

   `= 1`

   `=`ডানপক্ষ

  `:. (x^a/x^b)^(1/(ab)) . (x^b/x^c)^(1/(bc)) . (x^c/x^a)^(1/(ca))`

  ` = 1`  (প্রমাণিত)

Question:১৬. প্রমাণ কর: `(x^a/x^b)^(a + b) . (x^b/x^c)^(b + c) . (x^c/x^a)^(c + a) = 1` 

সমাধান: 
 বামপক্ষ 
 `= (x^a/x^b)^(a + b) . (x^b/x^c)^(b + c) . (x^c/x^a)^(c + a)`

 `= (x^(a - b))^(a + b) . (x^(b - c))^(b + c) . (x^(c - a))^(c + a)`

 `= x^((a - b)(a + b)) . x^((b - c)(b + c)) . x^((c - a)(c + a))`

 `= x^(a^2 - b^2) . x^(b^2 - c^2) . x^(c^2 - a^2)`

 `= x^(a^2 - b^2 + b^2 - c^2 + c^2 - a^2)`

 `=x^0`

 `= 1`

 `=` ডানপক্ষ

 `:. (x^a/x^b)^(a + b) . (x^b/x^c)^(b + c) . (x^c/x^a)^(c + a) `

      `= 1`  (প্রমাণিত)

Question:১৭. প্রমাণ কর:`(x^p/x^q)^(p + q - r) xx (x^q/x^r)^(q + r - p) xx (x^r/x^p)^(r + p - q) = 1` 

সমাধান: 
  বামপক্ষ 
  `= (x^p/x^q)^(p + q - r) xx (x^q/x^r)^(q + r - p) xx (x^r/x^p)^(r + p - q)`

  `= (x^(p - q))^(p + q - r) xx (x^(q - r))^(q + r - p) xx (x^(r - p))^(r + p - q)`

  `= x^((p - q)(p + q - r)) xx x^((q - r)(q + r - p)) xx x^((r - p)(r + p - q))`

  `= x^((p - q)(p + q) - (p - q)r) xx x^((q - r)(q + r) - (q - r)p) xx x^((r - p)(r + p) - (r - p)q)`

  `= x^(p^2 - q^2 - pr + qr) xx x^(q^2 - r^2 - pq + pr) xx x^(r^2 - p^2 - qr + pq)`

  `= x^(p^2 - q^2 - pr + qr + q^2 - r^2 - pq + pr + r^2 - p^2 - qr + pq)`

  `= x^0`

  `= 1`

  `=`ডানপক্ষ

`:. (x^p/x^q)^(p + q - r) xx (x^q/x^r)^(q + r - p) xx (x^r/x^p)^(r + p - q)`

   ` = 1`   (প্রমাণিত)