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The given expression is:

\[
\left( 10x^2 y \right)^{\frac{3}{11}}
\]

To convert this expression into radical form, we need to understand the fractional exponent \(\frac{3}{11}\). The denominator of the fraction (11) corresponds to the index of the radical, while the numerator (3) represents the power to which the entire expression is raised.

The general rule for converting a fractional exponent to radical form is:

\[
a^{\frac{m}{n}} = \sqrt[n]{a^m}
\]

Applying this rule to the expression:

\[
\left( 10x^2 y \right)^{\frac{3}{11}} = \sqrt[11]{(10x^2 y)^3}
\]

Now, let's expand \((10x^2 y)^3\):

\[
(10x^2 y)^3 = 10^3 \cdot (x^2)^3 \cdot y^3 = 1000 \cdot x^6 \cdot y^3
\]

Thus, the expression in radical form becomes:

\[
\sqrt[11]{1000 \cdot x^6 \cdot y^3}
\]

This is the radical form of the given expression.

Would you like more details or have any other questions?

Here are five related questions:
1. How would the expression change if the exponent were \(\frac{5}{7}\) instead of \(\frac{3}{11}\)?
2. What is the general rule for converting a power of a product into a product of powers?
3. How would you simplify the radical if the base contained negative values?
4. Can you write \((10x^2 y)^{\frac{3}{11}}\) as a product of smaller radicals?
5. How does fractional exponentiation relate to logarithms?

**Tip**: When working with fractional exponents, always remember that the denominator indicates the root, while the numerator indicates the power.

Write the expression in radical form. Assume that all variables represent positive real numbers: (10x^2 y)^(3/11).

Learn how to convert the expression (10x^2 y)^(3/11) into radical form. This problem covers key algebraic concepts like radicals and fractional exponents, using the formula a^(m/n) = n-th root of (a^m). Suitable for students in Grades 9-12.

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