Question

Solve:

`sqrt((2/5)^(4x - 3)) = 15 5/8`

Answer 1

Given:

`sqrt((2/5)^(4x - 3)) = 15 5/8`

Step-wise calculation:

1. Convert the mixed number right side to improper fraction:

`15 5/8 = (8 xx 15 + 5)/8`

`15 5/8 = (120 + 5)/8`

`15 5/8 = 125/8`

2. Square both sides to remove the square root on the left side:

`(sqrt((2/5)^(4x - 3)))^2 = (125/8)^2`

⇒ `(2/5)^(4x - 3) = 125^2/8^2`

⇒ `(2/5)^(4x - 3) = 15625/64`

3. Note that 125 = 5³ and 8 = 2³, so write the right-hand side as powers of 5 and 2:

`125^2/8^2 = (5^3)^2/((2^3)^2`

`125^2/8^2 = 5^6/2^6`

4. Rewrite the left-hand side base and the right side for comparison:

`(2/5)^(4x - 3) = (5/2)^(-(4x - 3)`

`(2/5)^(4x - 3) = 5^6/2^6`

This implies:

`(5/2)^(-(4x - 3)) = (5/2)^6`

5. Since bases are equal, set the exponents equal: 

–(4x – 3) = 6

6. Solve for (x):

–4x + 3 = 6

–4x = 6 – 3

–4x = 3

`x = -3/4`