Indices Questions and Answers

Indices Questions

1. Evaluate the value of x in

81 + 3 = 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 in the form A/B

where A and B are integers 5 x 3³

4. Find the value of x given that :

 2ˣ=0.0625 (x is an integer)

6. Find the value of x which satisfies the equation 16^x2 = 8 ^{4x – 3}

7. Solve the equation;

 9 ^x+1 + 3 ^{2x + 1}= 36

8. By letting P = 4^-y in the equation:

 4^-2y
+1 – 3 x 4^-y – 10 = 0

 (a) Write the above equation in terms of P

 (b) Hence find the possible values of y

9. Solve for x in the equation.

10. In the expansion of ax – 2
the constant term is 4860. Find the value of a.

x²

Indices Answers

1. 3⁴ x 3^4x + 3^4x = 246

 3^4x (81 + 1 ) = 246

82 x 3^4x = 246

82 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 = 3 y = log 3

log 5 = 0.6826

3. CBD = 90 – 42 = 48^o

Angle of triangle add to 180^o

DOB = 180^o – 42 = 138^o

Opposite angles of cyclic quadrilateral add to 180^o

DAB = 138^o = 69^o

2

Angle at circumference is half the nagle substended at centre by same chord

CDA

ABD= 90 – 48 = 42o

ADB = 180 –(69+42)

180-111=69^o

CDA = 90 + 69^o = 159^o

Show ADB is asoccesters

DAB = 69^o

DAB = 69^o

ADB = 69^o

ABD = 42^o

So two angles are equal hence it is asoccesters

4. 25 ^¾ = ( 25 ^½ ) ^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 = 3

5 x 25 125

5. 2^x = 0.0625 = 625

1000

2x = 1 = 2^-4

16

x = -4

6. 16x² = 8 ^4x-3

2^4×2 = 2 ^{3(4x -3)}

= 4^x2 = 12 x -9

= 4^x2 – 12x + 9 = 0

(2x-3)² = 0

2x-3 = 0

x = 1.5

7. 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

8. (a) 4p² – 3P – 10 =0

(b) 4p² – 8p + 5p =0

(4p +5) (p-2) = 0

p₁ = ^-5/₄, p =2

When y = ^-5/₄,

4^-y = -5

4

y = log ₄ (-5)

2

P = 2

4^-y = 2

2^-2y = 2¹

y = ^-1/₂

9.

1 = 1

16^x 32

1 = 1

2^4x 2⁵

2 + x = 2

-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

10. 15 (ax)⁴ (^-2/_x²) = 4860

60a⁴ = 4860

 a⁴ = 81

 a = 3