Math Problem Statement

express 108 1/3 in simplest radical form

Solution

To express 10813108 \frac{1}{3} in its simplest radical form, you are dealing with the cube root of 108, since 13\frac{1}{3} indicates a cube root:

10813=1083108^{\frac{1}{3}} = \sqrt[3]{108}

Step-by-Step Simplification:

  1. Prime factorization of 108:

108=22×33108 = 2^2 \times 3^3

  1. Simplify the cube root:

[ \sqrt[3]{108} = \sqrt[3]{2^2 \times 3^3} ]

  1. You can separate the cube root into:

[ \sqrt[3]{108} = \sqrt[3]{2^2} \times \sqrt[3]{3^3} ]

  1. Now, simplify further:
    • 333=3\sqrt[3]{3^3} = 3
    • 223\sqrt[3]{2^2} remains as 43\sqrt[3]{4}.

So, the simplest radical form is:

[ \sqrt[3]{108} = 3 \times \sqrt[3]{4} ]

Thus, 10813=343108^{\frac{1}{3}} = 3 \sqrt[3]{4}.

Let me know if you'd like further clarification or a detailed explanation!


Here are 5 follow-up questions to expand your understanding:

  1. How do you calculate cube roots for larger numbers?
  2. What is the difference between cube roots and square roots in simplification?
  3. Can you express 10823108 \frac{2}{3} in simplest radical form?
  4. How would you simplify 33233 \sqrt[3]{32}?
  5. What role does prime factorization play in simplifying radical expressions?

Tip: Always start by breaking the number into its prime factors when simplifying radicals, especially for higher roots like cube roots.

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Math Problem Analysis

Mathematical Concepts

Radical Expressions
Cube Roots
Prime Factorization

Formulas

\( a^{\frac{1}{3}} = \sqrt[3]{a} \)
Prime factorization: \( 108 = 2^2 \times 3^3 \)

Theorems

Properties of Cube Roots

Suitable Grade Level

Grades 9-11