Math Problem Statement

Simplify the cube root of 343, \( \sqrt[3]{343} \).

Solution

To simplify 3433\sqrt[3]{343}, we need to determine whether 343 has any perfect cube factors.

  1. Prime Factorization of 343: 343=7×7×7=73343 = 7 \times 7 \times 7 = 7^3

  2. Cube Root Simplification: [ \sqrt[3]{343} = \sqrt[3]{7^3} ]

  3. The cube root of 737^3 simplifies to: 3433=7\sqrt[3]{343} = 7

Thus, the simplified form of 3433\sqrt[3]{343} is 77.

Would you like further details or have any other questions?

Here are some related questions to expand your understanding:

  1. What is the process for simplifying cube roots of larger numbers?
  2. How do cube roots differ from square roots in terms of simplification?
  3. What are the cube roots of other powers of 7, like 767^6?
  4. Can you explain how to estimate cube roots for non-perfect cubes?
  5. How do prime factorizations help in simplifying roots?

Tip: Always check for perfect cubes when simplifying cube roots.

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Math Problem Analysis

Mathematical Concepts

Simplification
Roots
Cube Roots

Formulas

\( \sqrt[3]{a^3} = a \)

Theorems

Properties of Exponents and Roots

Suitable Grade Level

Grades 8-10