Math Problem Statement

Simplify the expression \(\sqrt{4 \cdot \sqrt{6 \cdot \sqrt{36}}}\)

Solution

The expression you provided is:

4636\sqrt{4 \cdot \sqrt{6 \cdot \sqrt{36}}}

Let's break it down step-by-step:

Step 1: Simplify the innermost square root.

36=6\sqrt{36} = 6

Now the expression becomes: 466\sqrt{4 \cdot \sqrt{6 \cdot 6}}

Step 2: Multiply inside the second square root.

66=366 \cdot 6 = 36

Now the expression is: 436\sqrt{4 \cdot \sqrt{36}}

Step 3: Simplify the second square root.

36=6\sqrt{36} = 6

Now the expression becomes: 46\sqrt{4 \cdot 6}

Step 4: Multiply inside the square root.

46=244 \cdot 6 = 24

So the expression becomes: 24\sqrt{24}

Step 5: Simplify the square root.

24=26\sqrt{24} = 2\sqrt{6}

Thus, the final simplified result is: 262\sqrt{6}

Let me know if you want further details or have any questions!


Here are five related questions you might find helpful:

  1. How do you simplify nested square roots?
  2. What are the properties of square roots?
  3. How do you simplify ab\sqrt{a \cdot b}?
  4. How do you express a square root in its simplest radical form?
  5. How do you handle square roots of perfect squares?

Tip: When dealing with nested square roots, always start from the innermost root and work your way outward.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplification of Square Roots
Radical Expressions

Formulas

\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)

Theorems

Square Root Properties
Radical Simplification Rules

Suitable Grade Level

Grades 8-10