Express in simplest radical form.
$\left(125 x^6
ight)^{\frac{4}{3}}$
We are asked to simplify the expression \(\left(125 x^6\right)^{\frac{4}{3}}\). To do this, we will use fundamental properties of exponents. Specifically, we will use the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n = a^{m \times n}\). We also note that \(125\) can be expressed as a power of \(5\), since \(5 \times 5 \times 5 = 125\), which means \(125 = 5^3\).
First, we apply the property \((ab)^n = a^n b^n\) to the expression \(\left(125 x^6\right)^{\frac{4}{3}}\):
\[\left(125 x^6\right)^{\frac{4}{3}} = 125^{\frac{4}{3}} \times (x^6)^{\frac{4}{3}}.\]
Next, substitute \(5^3\) for \(125\):
\[\left(5^3\right)^{\frac{4}{3}} \times (x^6)^{\frac{4}{3}}.\]
Now, apply the power of a power rule \((a^m)^n = a^{m \times n}\) to each factor separately:
For the first factor:
\[\left(5^3\right)^{\frac{4}{3}} = 5^{3 \times \frac{4}{3}} = 5^4.\]
For the second factor:
\[(x^6)^{\frac{4}{3}} = x^{6 \times \frac{4}{3}} = x^8.\]
Finally, we compute the numerical value for the first factor:
\[5^4 = 5 \times 5 \times 5 \times 5 = 625.\]
Combining the two simplified parts, we get:
\[625 \times x^8.\]
Thus, the expression \(\left(125 x^6\right)^{\frac{4}{3}}\) simplifies to \(625 x^8\).
In engineering and physics, exponents are frequently used to handle scaling factors. For instance, if the volume of a object like a sphere or a cube scales with a dimension \(x\) as \(V \propto x^3\), and you need to perform an operation involving the \(4/3\) power of that volume (perhaps in fluid dynamics or heat transfer analysis involving specific boundary layer thicknesses), you would arrive at an expression similar to the one solved. Simplifying these expressions allows engineers to quickly determine how a physical property changes with respect to a base measurement.