Note: `5^1∕4 xx (125)^0.25` `= 5^0.25 xx (5^3)^0.25` `= 5^0.25 xx 5^(3 xx 0.25)` `= 5^0.25 xx 5^0.75` `= 5^(0.25 + 0.75)` `= 5^1` `= 5`.

Note: `(32/243)^(-4∕5)` `= (243/32)^(4∕5)` `= (3^5/2^5)^(4∕5)` `= [(3/2)^5]^(4∕5)` `= (3/2)^(5 xx (4/5)` `= (3/2)^4` `= (3^5/2^5)^(4∕5)` `= [(3/2)^5]^(4∕5)` `= (3/2)^4` `= 3^4/2^4` `= 81/16`

Note: `(1/216)^(-2∕3) ÷ (1/27)^(-4∕3)` `= (216)^(2∕3) ÷ (27)^(4∕3)` `= (6^3)^(2∕3) ÷ (3^3)^(4∕3)` `= 6^(3 xx 2/3) ÷ 3^(3 xx 4/3)` `= 6^2 ÷ 3^4` `= 36/81` `= 4/9`

Note: Given Exp.` = (2^n . 2^4 - 2. 2^n)/(2 . 2^n . 2^3) + 1/2^3` `= (2^n (2^4 - 2))/(2^n (2 . 2^3)) + 1/2^3` `= ((16 - 2)/16 + 1/8)` `= (7/8 + 1/8)` `= 1`

Note: `(5sqrt(5) xx 5^3)/(5^(-3∕2)` `= 5^(a + 2) <=> (5^(3∕2) xx 5^3)/(5^(-3∕2)` `= 5^(a + 2)` `:. 5(3/2 + 3 + 3/2)` `= 5^(a + 2) or 5^6` `= 5^(a + 2)` `:. a + 2 = 6 or a = 4`

Note: `sqrt(2^n) = 64 <=> 2^(n∕2)` `= 64 = 2^6` `:. n/2 = 6 or n = 12.`

Note: Given Exp. `= (1 - (1/10)^-1)/((2^3/3) . (3/3)^3 + (-3)^1` `= (1 - 10)/(2^3/3 . 3^3/2^3 - 3)` `= -9/(9 - 3)` `= -9/6` `= -3/2.`

Note: `(9^n xx 3^5 xx (27)^3)/(3 xx (81)^4)` `= 27 => (3^2n xx 3^5 xx (3^3)^3)/(3 xx (3^4)^4` `= 3^3` `= (3^2n xx 3^5 xx 3^9)/(3^1 xx 3^16)` `= 3^3 or 3^(2n + 5 + 9)` `= 3^3 xx 3^1 xx 3^16 or 3^(2n + 14)` `= 3^20 `:. 2n + 14` `= 20 or 2n` `= 6 or n = 3`

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