Which expression is equivalent to $\left(x \cdot x^4 \cdot x^3\right)$?
We are given the expression \((x \cdot x^4)^{1/3}\). To simplify this, we need to apply the fundamental laws of exponents. The expression involves the product of powers with the same base, as well as a power of a power. The laws we will use are:
These rules allow us to combine the terms into a single, simplified form.
First, we focus on the base part of the expression inside the parentheses: \(x \cdot x^4\). Since \(x\) is the same as \(x^1\), we use the product of powers rule:
\[x^1 \cdot x^4 = x^{1+4} = x^5\]
Now, substituting this result back into our original expression, we get:
\[(x^5)^{1/3}\]
Next, we apply the power of a power rule, which states that \((x^a)^b = x^{a \cdot b}\). Here, \(a = 5\) and \(b = 1/3\). Multiplying these exponents together gives:
\[5 \cdot \left(\frac{1}{3}\right) = \frac{5}{3}\]
Thus, the simplified form of the entire expression is:
\[x^{5/3}\]
Comparing our simplified result \(x^{5/3}\) with the given choices:
The expression is equivalent to the second choice, \(x^{5/3}\). Therefore, the correct equivalent expression is \(\boxed{x^{5/3}}\).
The rules of exponents are vital for simplifying complex algebraic expressions, which often appear in scientific and engineering calculations. For instance, when calculating the growth rate of a population or the signal decay in a fiber optic cable, you might encounter nested powers that need to be reduced for easier computation. Mastering these rules allows you to work more efficiently with formulas involving scaling factors or dimensions.