Simplify the radical. Assume that the variable represents a positive real number.
$\sqrt{\frac{y^{19}}{y}}=$
We are tasked with simplifying the radical expression \(\sqrt{\frac{y^{19}}{y}}\) under the condition that the variable \(y\) is positive (\(y > 0\)). The expression involves a quotient of powers and a square root operation. By following the laws of exponents and radicals, we can reduce this expression to a much simpler form.
First, we look at the term inside the square root, which is \(\frac{y^{19}}{y}\). According to the quotient rule for exponents, \(\frac{y^m}{y^n} = y^{m-n}\) for \(y
eq 0\). Applying this to our expression:
\[\frac{y^{19}}{y^1} = y^{19-1} = y^{18}.\]
Thus, the expression becomes \(\sqrt{y^{18}}\).
Next, we simplify the square root of \(y^{18}\). Using the rule that \(\sqrt{x^k} = (x^k)^{1/2} = x^{k/2}\), we have:
\[\sqrt{y^{18}} = (y^{18})^{1/2} = y^{18 \times \frac{1}{2}} = y^9.\]
Since we are given that \(y > 0\), the value \(y^9\) is positive, and no additional absolute value signs are necessary. Therefore, the simplified expression is \(y^9\).
Simplifying radicals and exponents is fundamental in many scientific and engineering fields. For example, when calculating the surface area or volume of geometric shapes like a sphere, where the radius might be given in terms of a variable, you often encounter complex powers and roots. Being able to simplify \(\sqrt{\frac{y^{19}}{y}}\) to \(y^9\) allows for much faster and cleaner calculations, similar to how an engineer might simplify a drag coefficient formula or a physicist might reduce an equation for particle energy to a more manageable state for further analysis.