USING PROPERTIES OF EQUALITY TO SOLVE EQUATIONS WITH RADICALS

Solve the equation.

Problem 1 :

8^{x – 2} = √8

Solution :

8^{x – 2} = √8

8^{x – 2} = (8)^1/2

x – 2 = 1/2

x = 1/2 + 2

x = (1 + 4)/2

x = 5/2

Problem 2 :

√5 = 5^{x + 4}

Solution :

√5 = 5^{x + 4}

(5)^1/2 = 5^{x + 4}

1/2 = x + 4

1/2 – 4 = x

(1 – 8)/2 = x

-7/2 = x

Problem 3 :

(5^th√7)^x = 7^{2x + 3}

Solution :

(5^th√7)^x = 7^{2x + 3}

(7^1/5)^x = 7^{2x + 3}

(7)^x/5 = 7^{2x + 3}

x/5 = 2x + 3

Multiplying 5 on each sides.

(x/5) × 5 = (2x + 3) × 5

x = 10x + 15

x – 10x = 15

-9x = 15

x = -15/9

x = -5/3

Problem 4 :

12^{2x – 1} = (∜12)^x

Solution :

12^{2x – 1} = (∜12)^x

12^{2x – 1} = (12^1/4)^x

12^{2x – 1} = (12)^x/4

2x – 1 = x/4

Multiplying 4 on each sides.

(2x – 1) × 4 = x/4 × 4

8x – 4 = x

8x – x = 4

7x = 4

x = 4/7

Problem 5 :

(∛6)^2x = (√6)^{x + 6}

Solution :

(∛6)^2x = (√6)^{x + 6}

(6^1/3)^2x = (6^1/2)^{x + 6}

6^2x/3 = 6^{(x + 6)/2}

6^2x/3 = 6^{x/2 + 3}

2x/3 = x/2 + 3

LCM (3, 2) = 6

(2x/3 × 2/2) = (x/2 × 3/3) + (3/1 × 6/6)

4x/6 = 3x/6 + 18/6

4x/6 – 3x/6 = 18/6

x/6 = 18/6

x/6 = 3

Multiplying 6 on each sides.

x = 18

Problem 6 :

(5^th√3)^{5x - 10} = (8^th√3)^4x

Solution :

(5^th√3)^{5x - 10} = (8^th√3)^4x

(3^1/5)^{5x - 10} = (3^1/8)^4x

(3)^{(5x – 10)/5} = (3)^4x/8

(3)^{(x – 2)} = 3^x/2

x – 2 = x/2

x – x/2 = 2

(2x – x)/2 = 2

x/2 = 2

Multiplying 2 on each sides.

x = 4