Math Problem Statement
express 108 1/3 in simplest radical form
Solution
To express in its simplest radical form, you are dealing with the cube root of 108, since indicates a cube root:
Step-by-Step Simplification:
- Prime factorization of 108:
- Simplify the cube root:
[ \sqrt[3]{108} = \sqrt[3]{2^2 \times 3^3} ]
- You can separate the cube root into:
[ \sqrt[3]{108} = \sqrt[3]{2^2} \times \sqrt[3]{3^3} ]
- Now, simplify further:
- remains as .
So, the simplest radical form is:
[ \sqrt[3]{108} = 3 \times \sqrt[3]{4} ]
Thus, .
Let me know if you'd like further clarification or a detailed explanation!
Here are 5 follow-up questions to expand your understanding:
- How do you calculate cube roots for larger numbers?
- What is the difference between cube roots and square roots in simplification?
- Can you express in simplest radical form?
- How would you simplify ?
- 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