Verified Answer :

  • Factor the coefficient \(81\) into \(27 \times 3\), where \(27\) is a perfect cube \(3^3\).
  • Factor the variable \(a^8\) into \(a^6 \times a^2\), where \(a^6\) is a perfect cube \((a^2)^3\).
  • Group the perfect cube terms: \(\sqrt[3]{(27 \cdot a^6) \cdot (3 \cdot a^2)}\).
  • Extract the cube roots to get the final simplified form \(\boxed{3 a^2 \sqrt[3]{3 a^2}}\).

Explanation

  1. Analyzing the Expression

The goal is to simplify the algebraic expression \(\sqrt[3]{81 a^8}\). To simplify a cube root, we look for factors inside the radical that are perfect cubes. A perfect cube is an expression that can be written in the form \(x^3\). We need to factor the constant \(81\) and the variable part \(a^8\) into perfect cube components and remaining terms.

  1. Factorization and Simplification

First, consider the constant part, \(81\). We know that \(27\) is a perfect cube because \(27 = 3^3\). Since \(81 = 27 \times 3\), we can write the cube root of 81 as \(\sqrt[3]{27 \cdot 3} = 3\sqrt[3]{3}\). Next, consider the variable part, \(a^8\). To extract the largest perfect cube factor, we use the laws of exponents, specifically \((x^m)^n = x^{mn}\). We can rewrite \(a^8\) as \(a^6 \cdot a^2\) because \(a^6 = (a^2)^3\). Thus, \(\sqrt[3]{a^8} = \sqrt[3]{a^6 \cdot a^2} = \sqrt[3]{(a^2)^3 \cdot a^2} = a^2 \sqrt[3]{a^2}\).

  1. Final Simplification

Now, combine these two parts inside the cube root:
\[\sqrt[3]{81 a^8} = \sqrt[3]{(27 \cdot 3) \cdot (a^6 \cdot a^2)} = \sqrt[3]{(27 \cdot a^6) \cdot (3 \cdot a^2)}\]
Using the radical property \(\sqrt[n]{xy} = \sqrt[n]{x} \cdot \sqrt[n]{y}\), we separate the terms:
\[\sqrt[3]{27 a^6} \cdot \sqrt[3]{3 a^2}\]
Since \(27 = 3^3\) and \(a^6 = (a^2)^3\), we have \(\sqrt[3]{27 a^6} = \sqrt[3]{(3 a^2)^3} = 3 a^2\).
Therefore, the full expression simplifies to:
\[3 a^2 \sqrt[3]{3 a^2}\]

  1. Conclusion

Comparing our result \(3 a^2 \sqrt[3]{3 a^2}\) with the given options, we identify that the first option is the correct one. Therefore, the simplified expression is \(\boxed{3 a^2 \sqrt[3]{3 a^2}}\).

Examples

Simplifying radicals is a fundamental skill in geometry and physics. For instance, if you are designing a storage box in the shape of a cube and you know the volume \(V = 81 a^8\) cubic units, finding the edge length of the box requires taking the cube root of the volume. By simplifying \(\sqrt[3]{81 a^8}\) to \(3 a^2 \sqrt[3]{3 a^2}\), you can easily estimate the dimensions of the box for any value of \(a\), making your design process much more efficient.

Related Questions