Simplify.
$-2 \sqrt[4]{162}-\sqrt[4]{512}$
We aim to simplify the expression \(-2 \sqrt[4]{162} - \sqrt[4]{512}\). To do this, we need to decompose each number under the radical into factors that are perfect fourth powers, since we are working with fourth roots. A fourth root \(\sqrt[4]{x}\) can be simplified if \(x\) contains a factor of the form \(a^4\), because \(\sqrt[4]{a^4 \cdot b} = a \sqrt[4]{b}\).
First, let's factorize \(162\). We can see that \(162 = 2 \times 81 = 2 \times 3^4\). Applying the property of radicals, we get:
\[\sqrt[4]{162} = \sqrt[4]{3^4 \cdot 2} = 3 \sqrt[4]{2}.\]
Next, let's factorize \(512\). We know that \(512 = 2^9\). To extract the largest fourth power, we write \(512 = 2^8 \cdot 2\). Since \(2^8 = (2^2)^4 = 4^4 = 256\), we have:
\[\sqrt[4]{512} = \sqrt[4]{2^8 \cdot 2} = \sqrt[4]{(2^2)^4 \cdot 2} = 2^2 \sqrt[4]{2} = 4 \sqrt[4]{2}.\]
Now, substitute the simplified forms back into the original expression \(-2 \sqrt[4]{162} - \sqrt[4]{512}\):
\[-2(3 \sqrt[4]{2}) - 4 \sqrt[4]{2}.\]
Distribute the \(-2\):
\[-6 \sqrt[4]{2} - 4 \sqrt[4]{2}.\]
Combining the like terms, we add the coefficients:
\[(-6 - 4) \sqrt[4]{2} = -10 \sqrt[4]{2}.\]
Thus, the simplified expression is \(-10 \sqrt[4]{2}\).
This type of simplification is fundamental in algebra and physics, such as when simplifying expressions involving dimensional analysis or calculating exact values for radical expressions. For instance, in engineering, when working with electrical circuits, you might encounter radical expressions for impedances; simplifying them helps in comparing different circuit configurations quickly and accurately, preventing rounding errors that occur with decimal approximations.