Math Problem Statement

Simplify the expression \(\sqrt[3]{4} \cdot \sqrt[3]{9}\).

Solution

This expression involves multiplying two cube roots:

[ \sqrt[3]{4} \cdot \sqrt[3]{9} ]

To simplify, we can use the property of radicals:

[ \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} ]

So:

[ \sqrt[3]{4} \cdot \sqrt[3]{9} = \sqrt[3]{4 \cdot 9} ]

[ \sqrt[3]{4 \cdot 9} = \sqrt[3]{36} ]

The result is:

363\sqrt[3]{36}

This is the simplified form, as 363\sqrt[3]{36} cannot be simplified further unless we compute its approximate decimal value. Let me know if you'd like me to calculate that or explain further!


Questions to expand your understanding:

  1. What are the properties of cube roots and how do they differ from square roots?
  2. How would this expression change if it involved square roots instead of cube roots?
  3. Can the cube root of 36 be written as a fractional exponent, and what does that represent?
  4. How can you compute the approximate decimal value of 363\sqrt[3]{36}?
  5. What are some real-world applications of cube roots in mathematics or physics?

Tip:

Remember that cube roots can handle negative numbers (e.g., 83=2\sqrt[3]{-8} = -2), unlike square roots for real numbers.

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Math Problem Analysis

Mathematical Concepts

Radicals
Cube Roots
Properties of Exponents

Formulas

\(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\)

Theorems

Properties of nth roots

Suitable Grade Level

Grades 9-11