Write the sum of $7 \sqrt{28 x^6}+4 \sqrt{7 x^6}$ in simplest form, if $x
eq 0$.
To simplify the expression \(7 \sqrt{28 x^6} + 4 \sqrt{7 x^6}\) given \(x eq 0\), we first analyze each term individually using the properties of square roots. Recall that for any real numbers \(a \geq 0\) and \(b \geq 0\), \(\sqrt{ab} = \sqrt{a}\sqrt{b}\), and for any real number \(y\), \(\sqrt{y^2} = |y|\).
For the first term, \(7 \sqrt{28 x^6}\), we can factor the radicand:
\[ 28 x^6 = 4 \cdot 7 \cdot (x^3)^2 \]
Using the square root property:
\[ \sqrt{28 x^6} = \sqrt{4} \cdot \sqrt{7} \cdot \sqrt{(x^3)^2} = 2 \cdot \sqrt{7} \cdot |x^3| \]
Therefore, the first term becomes:
\[ 7 \cdot (2 \cdot \sqrt{7} \cdot |x^3|) = 14 \sqrt{7} |x^3| \]
For the second term, \(4 \sqrt{7 x^6}\), we can factor the radicand:
\[ 7 x^6 = 7 \cdot (x^3)^2 \]
Using the square root property:
\[ \sqrt{7 x^6} = \sqrt{7} \cdot \sqrt{(x^3)^2} = \sqrt{7} \cdot |x^3| \]
Therefore, the second term becomes:
\[ 4 \cdot (\sqrt{7} \cdot |x^3|) = 4 \sqrt{7} |x^3| \]
Now, we sum the two simplified terms:
\[ 14 \sqrt{7} |x^3| + 4 \sqrt{7} |x^3| \]
Factoring out the common term \(\sqrt{7} |x^3|\):
\[ (14 + 4) \sqrt{7} |x^3| = 18 \sqrt{7} |x^3| \]
Since \(|x^3| = |x|^3\), we can also express this as \(18 \sqrt{7} |x|^3\). Both forms are equivalent. Thus, the simplified form of the expression is \(\boxed{18 \sqrt{7} |x|^3}\).
This type of radical simplification is essential in engineering and physical sciences when dealing with formulas derived from energy calculations or geometric relationships where squared variables appear under roots. For instance, if you are calculating the total distance traveled by an object influenced by a square-rooted factor related to its displacement squared, simplifying terms before computation allows for more efficient analysis and reduces potential calculation errors. By recognizing patterns, you can consolidate complex measurements into a single, manageable expression.