Question

Solve:

`sqrt((4/3)^(1 - 3x)) = 2 10/27`

Answer 1

Given equation: `sqrt((4/3)^(1 - 3x)) = 2 10/27`

Stepwise calculation:

1. Convert the mixed fraction on the right side into an improper fraction:

`2 10/27 = (2 xx 27 + 10)/27`

`2 10/27 = (54 + 10)/27`

`2 10/27 = 64/27`

2. Rewrite the equation using this value:

`sqrt((4/3)^(1 - 3x)) = 64/27`

3. Square both sides to eliminate the square root:

`(4/3)^(1 - 3x) = (64/27)^2`

4. Simplify the right side:

`(64/27)^2 = 64^2/27^2`

`(64/27)^2 = 4096/729`

5. Express bases as powers of integers:

`4/3 = (2^2/3^1)`

`4096/729 = 2^12/3^6`

6. Rewrite the left side:

`(2^2/3)^(1 - 3x) = 2^(2(1 - 3x))/(3^(1 - 3x))`

`(2^2/3)^(1 - 3x) = 2^(2 - 6x)/3^(1 - 3x)`

7. Equate the powers of 2 and 3 on both sides:

`2^(2 - 6x) = 2^(12)` 

`3^(1 - 3x) = 3^(6)`

From powers of 2:

2 – 6x = 12

⇒ –6x = 10 

⇒ `x = -10/6`

⇒ `x = -5/3`

From powers of 3:

1 – 3x = 6 

⇒ –3x = 5 

⇒ `x = -5/3`

Both give consistent value of `x = -5/3`.