Math Problem Statement

Simplify \( \sqrt[3]{147} \).

Solution

The problem requires simplifying the cube root of 147, written as 1473\sqrt[3]{147}.

To simplify this expression, let’s analyze if 147 can be broken down into factors that include a perfect cube:

  1. The prime factorization of 147 is: 147=3×7×7=3×72147 = 3 \times 7 \times 7 = 3 \times 7^2

  2. Since none of the factors form a perfect cube (like 232^3, 333^3, 535^3, etc.), the expression 1473\sqrt[3]{147} cannot be simplified further in terms of whole numbers or integers.

  3. Therefore, the simplest form of this expression is: 1473\sqrt[3]{147}

So, the answer to enter is:

1473\sqrt[3]{147}

Would you like further explanation on cube roots or prime factorization?

Here are some additional related questions to consider:

  1. What are the general steps for simplifying cube roots?
  2. How do you find the prime factorization of a number?
  3. Can cube roots of numbers that aren’t perfect cubes be expressed as exact decimals?
  4. How would this simplification differ if it were a square root instead of a cube root?
  5. Are there any practical applications of cube roots in real-life situations?

Tip: When simplifying roots, always check if the number can be factored into powers matching the root (like cubes for cube roots) for easier simplification.

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Math Problem Analysis

Mathematical Concepts

Cube Root
Prime Factorization

Formulas

Cube Root Simplification \( \sqrt[3]{a} \)

Theorems

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Suitable Grade Level

Grades 8-10