Respuesta :
We want to simplify
[tex]\sqrt[4]{256}[/tex]To simplify, we need to find the prime factors of this number. Since it is an even number, let's start by dividing by 2.
[tex]\begin{gathered} \frac{256}{2}=128 \\ \Rightarrow256=128\times2 \end{gathered}[/tex]The result still is an even number, this let us to keep dividing by 2.
[tex]\begin{gathered} \frac{128}{2}=64 \\ \Rightarrow256=(64\times2)\times2=64\times2^2 \end{gathered}[/tex]If we keep going
[tex]\begin{gathered} \frac{64}{2}=32,\frac{32}{2}=16,\frac{16}{2}=8,\frac{8}{2}=4,\frac{4}{2}=2,\frac{2}{2}=1 \\ \Rightarrow256=(32\times2)\times2^2=(16\times2)\times2^3=(8\times2)\times2^4\ldots \\ \Rightarrow256=1\times2^8=2^8 \end{gathered}[/tex]With this final equality, we have the following information:
[tex]256=2^8[/tex]We can rewrite the root like this:
[tex]\sqrt[4]{256}=\sqrt[4]{2^8}[/tex]We can also rewrite this power of two like this(just using potency properties):
[tex]2^8=(2^2)^4=4^4[/tex]Finally rewriting our original question
[tex]\sqrt[4]{256}=\sqrt[4]{4^4}[/tex]The exponent and the root cancel, then we get our answer.
[tex]\sqrt[4]{256}=4[/tex]