Simplify the following. Be sure to show your process and thinking.
$\frac{\left(2 x^5\right)^3}{6 x^{-7}}$
We need to simplify the algebraic expression \(\frac{(2x^5)^3}{6x^{-7}}\), where \(x eq 0\). This involves applying several laws of exponents, including the power of a power rule, the power of a product rule, and the quotient rule for exponents. Our objective is to reduce this expression to a single term consisting of a numerical coefficient and a single power of \(x\).
First, we focus on the numerator, \((2x^5)^3\). According to the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((x^m)^n = x^{m \cdot n}\), we have:
\[(2x^5)^3 = 2^3 \cdot (x^5)^3 = 8 \cdot x^{5 \cdot 3} = 8x^{15}.\]
Now, substitute this back into the original fraction:
\[\frac{8x^{15}}{6x^{-7}}.\]
Next, we simplify the numerical coefficients and the power terms separately. The coefficient is \(\frac{8}{6}\), which simplifies to \(\frac{4}{3}\) by dividing both the numerator and denominator by their greatest common divisor, \(2\). For the powers of \(x\), we use the quotient rule \(\frac{x^a}{x^b} = x^{a-b}\):
\[\frac{x^{15}}{x^{-7}} = x^{15 - (-7)} = x^{15 + 7} = x^{22}.\]
Combining these, we get:
\[\frac{4}{3}x^{22}.\]
By simplifying the numerical and variable parts of the expression, we have arrived at the final result:
\[\frac{(2x^5)^3}{6x^{-7}} = \frac{4}{3}x^{22}.\]
This expression is now fully simplified. The final answer is \(\boxed{\frac{4}{3}x^{22}}\).
In computer science and data compression, expressions involving powers are frequently simplified to manage resource allocation and processing speeds. For instance, if a data processing algorithm has a complexity factor defined by \(\frac{(2x^5)^3}{6x^{-7}}\), simplifying it to \(\frac{4}{3}x^{22}\) makes it much easier to estimate the time or space required to run the code for a given input size \(x\). Algebraic simplification is a foundational skill for optimizing any system that relies on mathematical models.