Question

What value of x would make the expression below equal to 8? Photo provided! 20points!

Answer 1

For this case, we must find the value of "x" so that the given expression is equal to 8.

That is to say:

$(\sqrt [5] {8 ^ 3}) ^ x = 8$

We apply "ln" to both sides of the equation to remove the exponent variable:

$ln ((\sqrt [5] {8 ^ 3}) ^ x) = ln (8)\\xln (\sqrt [5] {8 ^ 3}) = ln (8)\\xln (\sqrt [5] {512}) = ln (8)$

We rewrite 512 as:

$512 = 32 * 16 = 2 ^ 5 * 16\\xln (\sqrt [5] {2 ^ 5 * 16}) = ln (8)$

By definition of power properties we have:

$\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}$

So:

$xln (2 \sqrt [5] {16}) = ln (8)$

We clear x:

$x = \frac {ln (8)} {ln (2 \sqrt [5] {16})}$

In decimal form, $x = 1.6$ periodic number

ANswer:

$x = \frac {ln (8)} {ln (2 \sqrt [5] {16})}\\x = 1.6\ periodic\ number$