Math Problem Statement

What is the simplified form of the expression \( \sqrt[6]{11^3} \)?

Solution

The expression in the image appears to be the sixth root of 11311^3, written as:

1136\sqrt[6]{11^3}

This can be simplified or rewritten in exponential form. In terms of exponents, this is equivalent to:

(113)16=1136=1112(11^3)^{\frac{1}{6}} = 11^{\frac{3}{6}} = 11^{\frac{1}{2}}

Thus, the simplified result is:

1112=1111^{\frac{1}{2}} = \sqrt{11}

This represents the square root of 11.

Would you like further clarification on this process or any additional details?

Here are five related questions to explore:

  1. What are the steps to simplify higher-order roots of powers?
  2. How can this expression be evaluated as a decimal approximation?
  3. What are the properties of exponents when simplifying roots?
  4. How would you handle an expression like 1156\sqrt[6]{11^5}?
  5. What is the difference between simplifying roots and evaluating them?

Tip: When simplifying radicals, remember that amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}} helps convert roots into fractional exponents for easier manipulation!

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Math Problem Analysis

Mathematical Concepts

Radicals
Exponents
Simplification

Formulas

\( \sqrt[n]{a^m} = a^{m/n} \)

Theorems

Properties of Exponents
Exponentiation Theorem

Suitable Grade Level

Grades 9-11