Question

simplify.
$-2sqrt4{512} - sqrt4{162}$

Explanation:

Step1: Simplify \(\sqrt[4]{512}\)

We know that \(512 = 2^9\), so \(\sqrt[4]{512}=\sqrt[4]{2^8\times2}=2^2\sqrt[4]{2} = 4\sqrt[4]{2}\) (because \(\sqrt[4]{2^8}=2^{8/4}=2^2 = 4\)). Then \(- 2\sqrt[4]{512}=-2\times4\sqrt[4]{2}=-8\sqrt[4]{2}\).

Step2: Simplify \(\sqrt[4]{162}\)

We know that \(162=81\times2 = 3^4\times2\), so \(\sqrt[4]{162}=\sqrt[4]{3^4\times2}=3\sqrt[4]{2}\).

Step3: Combine the two terms

Now we have \(-2\sqrt[4]{512}-\sqrt[4]{162}=-8\sqrt[4]{2}-3\sqrt[4]{2}\). Since the radicals are the same, we can combine the coefficients: \((-8 - 3)\sqrt[4]{2}=-11\sqrt[4]{2}\).

Answer:

\(-11\sqrt[4]{2}\)