Verified Answer :

  • Identify the rule for converting radicals to rational exponents: \(\sqrt[n]{A} = A^{\frac{1}{n}}\).
  • Rewrite \(\sqrt[3]{x^8}\) as \((x^8)^{\frac{1}{3}}\).
  • Use the power of a power rule \((x^a)^b = x^{a \cdot b}\) to simplify the expression to \(x^{8 \cdot \frac{1}{3}}\).
  • Express the final result as \(\boxed{x^{\frac{8}{3}}}\).

Explanation

  1. Analysis of the Expression

The expression to be rewritten is \(\sqrt[3]{x^8}\). The radical notation \(\sqrt[n]{A}\) represents the \(n\)-th root of \(A\), which is equivalent to raising \(A\) to the power of \(\frac{1}{n}\). In this case, \(A = x^8\) and \(n = 3\).

  1. Rewriting the Radical as a Fractional Exponent

Using the rule \(\sqrt[n]{A} = A^{\frac{1}{n}}\), we can rewrite the expression as:
\[ \sqrt[3]{x^8} = (x^8)^{\frac{1}{3}} \]

  1. Simplifying the Exponent

Now we apply the power of a power rule for exponents, which states that \((x^a)^b = x^{a \cdot b}\). Applying this to our expression:
\[ (x^8)^{\frac{1}{3}} = x^{8 \cdot \frac{1}{3}} \]
Multiplying the exponents, we get:
\[ x^{\frac{8}{3}} \]

  1. Conclusion

The expression \(\sqrt[3]{x^8}\) expressed with rational exponents is \(x^{\frac{8}{3}}\). Thus, the final form is \(\boxed{x^{\frac{8}{3}}}\).

Examples

In fields like physics or engineering, you often deal with complex formulas involving roots and powers. For example, when calculating the energy density of a field that scales with the cube root of a parameter raised to the 8th power, rewriting \(\sqrt[3]{x^8}\) as \(x^{8/3}\) makes it significantly easier to perform algebraic operations or differentiate the expression. Mastery of these exponent rules allows for efficient manipulation of these scientific formulas.

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