Question

How can the answer be 106/27? I have worked this problem every way imaginable.

Answer 1

Here's why that is the case.

Explanation:

Your first goal here is to get rid of the exponents, so start by raising both sides of the equation to the fourth power. You need to do that because if you multiply the two fractional exponents by `4`, you will end up with exponents equal to `1`.

`[3 * (4 - t)^(1/4)]^4 = (6^(1/4))^4`

This will get you

`3^4 * (4 - t)^((1/4 * 4)) = 6^((1/4 * 4))`

which simplifies to

`3^4 * (4-t) = 6`

At this point, all you have to do is to isolate `t`. Start by dividing both sides by `3^4`

`4 - t = 6/3^4`

This is equivalent to

`t - 4 = - 6/3^4`

Add `4` to both sides to get

`t = 4 - (2 * color(red)(cancel(color(black)(3))))/(3^3 * color(red)(cancel(color(black)(3))))`

`t = 4 - 2/27`

You can thus say that

`t = (27 * 4 - 2)/27 = 106/27`

Answer 2

Raise both sides to power of `4`. Expand, rearrange, simplify final fraction.

Explanation:

`3(4 - t)^{1/4} = 6^{1/4}`
`3^4 (4 - t) = 6`
`81 (4 - t) = 6`
`4 - t = \frac{6}{81}`
`-t = \frac{6}{81} - 4`
`t = -\frac{6}{81} + \frac{324}{81}`
`t = \frac{318}{81}`
`t = \frac{106}{27}`