Simplify.
$\sqrt[3]{16 u^8} \cdot \sqrt[3]{4 u^3}$
Assume that the variable represents a positive real number.
We are given the expression \(\sqrt[3]{16 u^8} \cdot \sqrt[3]{4 u^3}\), where \(u\) is a positive real number. To simplify this, we can use the property of radicals that allows us to combine the product of two roots of the same index. The property states that \(\sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{x \cdot y}\). Since both radicals have index \(n=3\), we can combine them into a single cube root expression.
Applying the property, we have:
\[ \sqrt[3]{16 u^8} \cdot \sqrt[3]{4 u^3} = \sqrt[3]{(16 u^8) \cdot (4 u^3)} \]
Now, we simplify the product inside the cube root, which is known as the radicand:
\[ (16 u^8) \cdot (4 u^3) = (16 \cdot 4) \cdot (u^8 \cdot u^3) \]
Using the multiplication of coefficients and exponent rules (\(a^m \cdot a^n = a^{m+n}\)), we get:
\[ 16 \cdot 4 = 64 \]
\[ u^8 \cdot u^3 = u^{8+3} = u^{11} \]
Thus, the expression becomes \(\sqrt[3]{64 u^{11}}\).
To simplify \(\sqrt[3]{64 u^{11}}\), we look for perfect cubes.
First, for the coefficient: \(64 = 4^3\), so \(\sqrt[3]{64} = 4\).
Next, for the variable part \(u^{11}\), we can rewrite it to isolate powers that are multiples of 3:
\[ u^{11} = u^9 \cdot u^2 = (u^3)^3 \cdot u^2 \]
Now substitute these back into the cube root:
\[ \sqrt[3]{64 u^{11}} = \sqrt[3]{4^3 \cdot (u^3)^3 \cdot u^2} \]
Using the property \(\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}\), we distribute the cube root:
\[ \sqrt[3]{4^3} \cdot \sqrt[3]{(u^3)^3} \cdot \sqrt[3]{u^2} = 4 \cdot u^3 \cdot \sqrt[3]{u^2} \]
Therefore, the simplified form is \(4 u^3 \sqrt[3]{u^2}\).
Imagine you are designing a storage box where the volume needs to be calculated in segments. If one part of a measurement involves a cubic expansion (like finding the side length of a volume \(V = 64 u^{11}\)), simplifying these expressions helps in architectural or industrial design. For example, if you know the volume of a cubic container is \(V = 64 u^{11}\), finding the side length \(s = \sqrt[3]{V}\) requires precisely this simplification: \(s = \sqrt[3]{64 u^{11}} = 4 u^3 \sqrt[3]{u^2}\). This allows for efficient planning of material dimensions based on variable input sizes.