1. If 5^(a + b) = 5 x 25 x 125, find the value of (a + b)^2 ?
Given 5^(a+b) = 5^1 x 5^2 x 5^3 => 5^(a+b) = 5^(1+2+3) => 5^(a + b) = 5^6
(a+b) = 6 => ( a + b )^2 = 36
2. Evaluate 1000^7 / 10^14 = ?
Correct Ans:10 7
1000^7 / 10^14 = (10^3)^7 x 10^–14 = 10^(21 – 14) = 10^7
3. If a = 1 / ( 3 – 2 ?2 ) then find 1/a =?
Correct Ans:3 + 2?2
Multiplying with conjugate we get a = (3 + 2?2)/1 = 3 + 2 ?2
4. If a and b are positive integers such that a^b = 121 then (a-1)^(b + 1) = ?
a^b = 121 => a^b=11^2=> a = 11 and b =2
so (11-1)^(2+1) = 10^3 = 1000
5. (256)^0.16 x (256)^0.09 = ?
(256)0.16 x (256)0.09 = (256)0.25 = 2561/4 = (44)1/4 = 4
(0.04)^–1.5 = (1/25)^ –1.5= 25^1.5 = 25^3/2 = (25)3/2 = 25xsqrt(25) = 25 x 5 = 125
7. 3^( x – y )=27 and 3^( x + y )=243, then x is equal to
3^( x – y) = 27 => 3 (x – y) = 3^3 => x – y = 3
3^( x + y) = 243 => 3^(x + y) = 3^5 => x + y = 5
Adding these two equations, we get 2x = 8 => x = 4
8. If 5^a = 3125, then the value of 5^(a-3) = ?
5a = 3125 => 5a = 55 => a=5 and 5a-3 = 55-3 = 52 = 25
9. (16)3.5 x (16)? = 2564
(16)^3.5 x 16^x = 2564 => 16^( x + 3.5) = (16^2)4 = 16^8 => x+ 3.5 = 8 => x = 4.5
10. If (a/b)^(x-1)=(b/a)^(x-3) then the value of x is:
(a/b)x-1 = (b/a)x-3 => (a/b)x-1 = (a/b)3-x => x-1 = 3-x => 2x = 4 => x=2
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