Verified Answer :

  • Combine the radicals into one using the property \(\sqrt[5]{a} \cdot \sqrt[5]{b} = \sqrt[5]{ab}\).
  • Multiply the coefficients and the variable powers inside the radical: \(\sqrt[5]{8u^3 \cdot 12u^2} = \sqrt[5]{96u^5}\).
  • Simplify the radical by extracting perfect fifth powers: \(\sqrt[5]{96u^5} = \sqrt[5]{32 \cdot 3} \cdot \sqrt[5]{u^5} = 2 \cdot \sqrt[5]{3} \cdot u\).
  • State the final simplified form as \(\boxed{2u \sqrt[5]{3}}\).

Explanation

  1. Combining the radicals

To simplify the expression \(\sqrt[5]{8 u^3} \cdot \sqrt[5]{12 u^2}\) for a positive real number \(u\), we first utilize the property of radicals that allows us to combine the product of two roots of the same order into a single root: \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\). Applying this rule, the expression becomes \(\sqrt[5]{(8 u^3) \cdot (12 u^2)}\).

  1. Multiplying terms inside the radical

Next, we perform the multiplication inside the radical. Multiply the coefficients \(8\) and \(12\) to get \(96\), and use the exponent rule \(u^m \cdot u^n = u^{m+n}\) to multiply the variable terms \(u^3\) and \(u^2\), resulting in \(u^5\). The expression now reads \(\sqrt[5]{96 u^5}\).

  1. Simplifying the radical components

We can now separate the radical into the product of two roots: \(\sqrt[5]{96} \cdot \sqrt[5]{u^5}\). Since \(u > 0\), \(\sqrt[5]{u^5} = u\). For the constant coefficient, observe that \(96 = 32 \cdot 3\), where \(32\) is a perfect fifth power since \(2^5 = 32\). Thus, \(\sqrt[5]{96} = \sqrt[5]{32} \cdot \sqrt[5]{3} = 2 \sqrt[5]{3}\).

  1. Final simplification

Finally, we combine the simplified components to arrive at the result: \(2 \cdot u \cdot \sqrt[5]{3} = 2u \sqrt[5]{3}\). The original expression \(\sqrt[5]{8 u^3} \cdot \sqrt[5]{12 u^2}\) simplifies to \(2u \sqrt[5]{3}\).

Examples

In fields like structural engineering or physics, radical expressions frequently appear when calculating properties related to volume, cross-sectional areas, or physical scaling laws. For example, if you are determining the scale factor for a 5-dimensional geometric object's volume where individual dimensions are expressed as roots, simplifying those expressions allows you to find the exact ratio needed for construction or material estimation. This ensures precision when dealing with complex multi-dimensional scaling problems.

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