Math Problem Statement

Go step by step to reduce the radical. \( \sqrt{192} \)

Solution

The given problem asks to simplify the square root of 192 step by step.

To reduce 192\sqrt{192}, we need to factor 192 into its prime factors and find the perfect squares.

Step-by-Step Simplification:

  1. Start by finding the prime factorization of 192.

    192÷2=96192 \div 2 = 96
    96÷2=4896 \div 2 = 48
    48÷2=2448 \div 2 = 24
    24÷2=1224 \div 2 = 12
    12÷2=612 \div 2 = 6
    6÷2=36 \div 2 = 3
    3÷3=13 \div 3 = 1

    So, the prime factorization of 192 is: 192=26×3192 = 2^6 \times 3

  2. Group the factors into perfect squares where possible: 192=(26)×3=(23)2×3192 = (2^6) \times 3 = (2^3)^2 \times 3

  3. Simplify the square root by taking out the perfect square: 192=(23)2×3=23×3=83\sqrt{192} = \sqrt{(2^3)^2 \times 3} = 2^3 \times \sqrt{3} = 8\sqrt{3}

Final Answer:

192=83\sqrt{192} = 8\sqrt{3}

Would you like more details on any of the steps or further clarification?

Related Questions:

  1. How do you factor other square roots into prime factors?
  2. What is the method to handle cube roots or higher roots similarly?
  3. How do you simplify 75\sqrt{75} using prime factorization?
  4. What happens when the number inside the radical is already a perfect square?
  5. How can you break down cube roots using prime factorization?

Tip:

Always check if the number inside the square root has any perfect square factors, as this will help simplify the radical easily!

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Math Problem Analysis

Mathematical Concepts

Radicals
Prime Factorization

Formulas

\( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)
Prime Factorization of 192: \( 192 = 2^6 \times 3 \)

Theorems

Square root of a product theorem

Suitable Grade Level

Grades 8-10