Indices Questions and Answers - Form 2 Topical Mathematics

1. Evaluate the value of x in
   81^(x+1) + 3^4x = 246
2. Solve for y in the equation:-
   5^(2y+1) = 4(5)^y+1 − 15
3. Without logarithm tables or calculators, evaluate: (25)^¾ x 0.92 x 22/_{5⁵⁄2 x 3³} in the form A/B where A and B are integers
4. Find the value of x given that :
   2^x=0.0625 (x is an integer)
5. Find the value of x which satisfies the equation 16^x2 = 8^{(4x – 3)}
6. Solve the equation;
   9^(x+1) + 3^(2x+1)= 36
7. By letting P = 4^-y in the equation:
   4^(-2y+1) − 3 x 4^-y − 10 = 0
   1. Write the above equation in terms of P
   2. Hence find the possible values of y
8. Solve for x in the equation
   ¹/₁₆^x = ¹/₃₂
9. In the expansion of [ax − ²/_x²]⁶ the constant term is 4860. Find the value of a.
   
   

ANSWERS

1. 3⁴ x 3^4x + 3^4x = 246
   3^4x (81 + 1 ) = 246
   ⁸²/₈₂ x 3^4x = ²⁴⁶/₈₂
   3^4x = 3¹    √
   4x = 1
   x = ¹/₄
2. 5^2y x5¹ = 4^(5y+1) − 15
   5^y x 5^y x 5¹ = 4 x 5y x 51 − 15
   Let 5y = t
   5t² = 20t − 15
   t² = 20t − 15
   t² − 4t + 3 = 0
   (t-1) (t-3) = 0
   t = 1 or 3
   5y = 1= 5^o
   Or 5y = 3y = log3
   log5 = 0.6826
3. 25¾ = (25½)3/2  = 5
   0.92 = (9/10)2 = 92/100
   22 = 22
   (√5)3 x 92 x 22 / (√5)5 x 102 x 33
   3 x 4/(√5)2 x 102
   3/(5 X 25) = 3/125
4. 2^x = 0.0625 = ⁶²⁵/₁₀₀₀
   2x = ¹/₁₆ = 2^-4
   x = -4
5. 16x² = 8^(4x-3)
   2^(4x2) = 2 ^{3(4x -3)}
   = 4^x2 = 12x -9
   = 4^x2 − 12x + 9 = 0
   (2x−3)² = 0
   2x−3 = 0
   x = 1.5
   
6.  9 ^{(x+ 1)} + 3^(2x+1) = 36

   3^(2x+2) + 3^{(2x+ 1)} = 36
   3^2x(9+3) = 36
   3^2x = 3¹
   2x = 1
   x = ½

7. 1. 4p² − 3P −10 =0
   2. 4p² − 8p + 5p =0
      (4p +5) (p−2) = 0
      p1 = ^-5/₄, p =2
      When y = ^-5/₄,
      4^-y = ^-5/₄
      y = ^{log₄ (-5)}/ ₂
      P = 2
      4^-y = 2
      2^-2y = 2¹
      y = ^-1/₂
8. ¹/₁₆^x = ¹/₃₂
   [¹/₂^4x ]^{(x- ¼)}= ¹/_2⁵
   
   2^{(-4xˆ2 + x)} + x = 2^-5
   -4x² + x + 5 = 0
   4x² − x − 5 = 0
   4x² −5x + 4x – 5 = 0
   x(4x – 5) + 1(4x – 5) = 0
   x = -1 or x = ⁵/₄
9. 15 (ax)⁴ (^-2/_x²) = 4860
   60a⁴ = 4860
   a⁴ = 81
   a = 3