For this case, we must find the value of "x" so that the given expression is equal to 8.
That is to say:
[tex](\sqrt [5] {8 ^ 3}) ^ x = 8[/tex]
We apply "ln" to both sides of the equation to remove the exponent variable:
[tex]ln ((\sqrt [5] {8 ^ 3}) ^ x) = ln (8)\\xln (\sqrt [5] {8 ^ 3}) = ln (8)\\xln (\sqrt [5] {512}) = ln (8)[/tex]
We rewrite 512 as:
[tex]512 = 32 * 16 = 2 ^ 5 * 16\\xln (\sqrt [5] {2 ^ 5 * 16}) = ln (8)[/tex]
By definition of power properties we have:
[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]
So:
[tex]xln (2 \sqrt [5] {16}) = ln (8)[/tex]
We clear x:
[tex]x = \frac {ln (8)} {ln (2 \sqrt [5] {16})}[/tex]
In decimal form, [tex]x = 1.6[/tex] periodic number
ANswer:
[tex]x = \frac {ln (8)} {ln (2 \sqrt [5] {16})}\\x = 1.6\ periodic\ number[/tex]
