Math Problem Statement

Simplify \( \sqrt{54} \). Show all your work.

Solution

To simplify 54\sqrt{54}, we break it down step by step:

Step 1: Prime factorization of 54

The number 5454 can be factored into: 54=2×3×3×3=2×3354 = 2 \times 3 \times 3 \times 3 = 2 \times 3^3

Step 2: Apply the square root

The square root of 5454 is: 54=2×33\sqrt{54} = \sqrt{2 \times 3^3}

We can split the square root as: 54=2×33\sqrt{54} = \sqrt{2} \times \sqrt{3^3}

Step 3: Simplify perfect squares

Within 33\sqrt{3^3}, the 323^2 part can be simplified: 33=32×3=32×3=33\sqrt{3^3} = \sqrt{3^2 \times 3} = \sqrt{3^2} \times \sqrt{3} = 3\sqrt{3}

Thus: 54=2×33=36\sqrt{54} = \sqrt{2} \times 3\sqrt{3} = 3\sqrt{6}

Final Answer:

54=36\sqrt{54} = 3\sqrt{6}

Would you like further details or have additional questions?

Related Questions:

  1. What is the process to simplify square roots for larger numbers?
  2. Can you explain how to identify perfect squares within a number?
  3. What is the difference between simplifying and approximating square roots?
  4. How do you handle square roots of decimals?
  5. Can this approach be used for cube roots?

Tip: Always break numbers into prime factors when simplifying square roots—it reveals perfect squares easily!

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Math Problem Analysis

Mathematical Concepts

Simplification of square roots
Prime factorization
Properties of square roots

Formulas

\( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)
\( \sqrt{n^2} = n \)

Theorems

Property of square roots (product rule)

Suitable Grade Level

Grades 6-8