Question

Solve for y:

`sqrt((2/5)^(y - 3)) = 39 1/16`

Answer 1

We are solving:

`sqrt((2/5)^(y - 3)) = 39 1/16`

Step 1: Convert mixed fraction

`39 1/16 = (39 * 16 + 1)/16`

= `(624 + 1)/16`

= `625/16`

So equation is:

`sqrt((2/5)^(y - 3)) = 625/16`

Step 2: Remove square root

`(2/5)^(y - 3) = (625/16)^2`

Step 3: Simplify right-hand side

625 = 25² = 5⁴,

16 = 2⁴

So, `(625/16)^2 = (5^4/2^4)^2 = (5/2)^8`

Step 4: Rewrite left-hand side

`(2/y)^(y - 3) = (5/2)^(-(y - 3)) = (5/2)^(3 - y)`

So equation is:

`(5/2)^(3 - y) = (5/2)^8`

Step 5: Equating exponents

3 – y = 8

y = 3 – 8 = –5