USING PROPERTY OF EQUALITY SOLVING EXPONENTIAL EQUATIONS

Solve each equations

Problem 1 :

16^-3r ⋅ 1/4 = 16

Solution :

16^-3r ⋅ 1/4 = 16

Multiplying 4 on both sides.

16^-3r ⋅ 1/4 × 4 = 16 × 4

16^-3r = 64

(2⁴)^-3r = 2⁶

2^-12r = 2⁶

By equating powers, we get

-12r = 6

Dividing -12 on both sides.

-12r/-12 = 6/-12

r = -1/2

So, the value of r is -1/2.

Problem 2 :

64^2x ⋅ 16^-3x = 1/16

Solution :

64^2x ⋅ 16^-3x = 1/16

64^2x ⋅ 16^-3x = 16^-1

(2⁶)^2x ⋅ (2⁴)^-3x = (2⁴)^-1

2^12x ⋅ 2^-12x = 2^-4

2^{12x – 12x} = 2^-4

No solution.

Problem 3 :

36^3x ⋅ (1/36)^2x = 6³

Solution :

36^3x ⋅ (1/36)^2x = 6³

36^3x ⋅ (36^-1) ^2x = 6³

(6²)^3x ⋅ ((6²)^-1)^2x = 6³

6^6x ⋅ (6^-2)^2x = 6³

6^6x ⋅ 6^-4x = 6³

6^{6x - 4x} = 6³

By equating powers, we get

6x – 4x = 3

2x = 3

Dividing 2 on both sides.

2x/2 = 3/2

x = 3/2

So, the value of x is 3/2.

Problem 4 :

243^-3r = 9^-2r

Solution :

243^-3r = 9^-2r

(3⁵)^-3r = (3²)^-2r

3^-15r = 3^-4r

By equating powers, we get

-15r = -4r

Adding 4r on both sides.

-15r + 4r = -4r + 4r

-11r = 0

r = 0

So, the value of r is 0.

Problem 5 :

(1/16)^-3n ⋅ 64^-2n = 1/16

Solution :

(1/16)^-3n ⋅ 64^-2n = 1/16

(16^-1)^-3n ⋅ 64^-2n = 16^-1

(16)³ⁿ ⋅ 64^-2n = 16^-1

(2⁴)³ⁿ ⋅ (2⁶)^-2n = (2⁴)^-1

2¹²ⁿ ⋅ 2^-12n = 2^-4

2^{12n - 12n} = 2^-4

2⁰ = 2^-4

So, no solution.

Problem 6 :

4^-p ⋅ 4^3p = 4³

Solution :

4^-p ⋅ 4^3p = 4³

4^{-p + 3p} = 4³

4^2p = 4³

By equating powers, we get

2p = 3

Dividing 2 on both sides.

2p/2 = 3/2

p = 3/2

So, the value of p is 3/2.

Problem 7 :

25^-x ⋅ 625 = 25

Solution :

25^-x ⋅ 625 = 25

25^-x ⋅ (25)² = 25

25^{-x + 2} = 25

-x + 2 = 1

Subtracting 2 on both sides.

-x + 2 – 2 = 1 – 2

-x = -1

x = 1

So, the value of x is 1.

Problem 8 :

36^3r = 216

Solution :

36^3r = 216

(6²)^3r = 6³

6^6r = 6³

By equating powers, we get

6r = 3

Dividing 6 on both sides.

6r/6 = 3/6

r = 1/2

So, the value of r is 1/2.

Problem 9 :

8^2m ⋅ 32^m = 16

Solution :

8^2m ⋅ 32^m = 16

(2³)^2m ⋅ (2⁵)^m = 2⁴

2^6m ⋅ 2^5m = 2⁴

2^{6m + 5m} = 2⁴

2^11m = 2⁴

By equating powers, we get

11m = 4

Dividing 11 on both sides.

11m/11 = 4/11

m = 4/11

So, the value of m is 4/11.

Problem 10 :

6^{2n + 2} = 36

Solution :

6^{2n + 2} = 36

6²ⁿ⁺² = 6²

By equating powers, we get

2n + 2 = 2

Subtracting 2 on both sides.

2n + 2 – 2 = 2

2n = 2

Dividing 2 on both sides.

2n/2 = 2/2

n = 1

So, the value of n is 1.

Problem 11 :

9^-v = 81

Solution :

9^-v = 81

9^-v = 9²

By equating powers, we get

-v = 2

v = -2

So, the value of v is -2.

Problem 12 :

6³ⁿ = 6³ⁿ

Solution :

6³ⁿ = 6³ⁿ

There are infinitely many solution.

Problem 13 :

A bacteria culture triples in size every hour. You begin observing the number of bacteria 2 hours after the culture is prepared. The amount y of bacteria x hours after the culture is prepared is represented by y = 162 (3^{x - 2}). When will there be 8100 bacteria ?

Solution :

y = 162 (3^{x - 2})

Here x represents number of hours and y represents number of bacteria.

When y = 8100, x = ?

8100 = 162 (3^{x - 2})

8100/162 = 3^{x - 2}

2700/54 = 3^{x - 2}

100/2 = 3^{x - 2}

50 = 3^{x - 2}

ln3 50 = x - 2

x = 5.5