Math Problem Statement

Evaluate \(\left( \sqrt{2 \times 3^2} \right)^3\)

Solution

The expression in the image appears to be:

(2×32)3\left( \sqrt{2 \times 3^2} \right)^3

To solve this, let's proceed with the steps:

  1. Simplify Inside the Square Root:

    The expression inside the square root is 2×322 \times 3^2.

    • Calculate 32=93^2 = 9.
    • Then, 2×9=182 \times 9 = 18.

    So, the expression becomes: (18)3\left( \sqrt{18} \right)^3

  2. Evaluate the Square Root:

    Find 18\sqrt{18}. 18=9×2=32\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}

    Now, we have: (32)3\left(3\sqrt{2}\right)^3

  3. Cube the Expression:

    Now, we cube 323\sqrt{2}: (32)3=33×(2)3(3\sqrt{2})^3 = 3^3 \times (\sqrt{2})^3

    • 33=273^3 = 27
    • (2)3=23/2=22(\sqrt{2})^3 = 2^{3/2} = 2 \sqrt{2}

    So, (32)3=27×22=542(3\sqrt{2})^3 = 27 \times 2 \sqrt{2} = 54 \sqrt{2}

Final Answer:

(2×32)3=542\left( \sqrt{2 \times 3^2} \right)^3 = 54 \sqrt{2}

Would you like further details on any step?


Related Questions

  1. How do you simplify square roots involving products?
  2. What is the process of cubing expressions with radicals?
  3. How do you express powers of square roots in simplified forms?
  4. Can you cube a square root directly, or are there simpler methods?
  5. How does exponentiation distribute over multiplication in expressions?

Tip

Always break down complex expressions into smaller, manageable steps to avoid calculation errors.

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Math Problem Analysis

Mathematical Concepts

Radicals
Exponentiation

Formulas

Square root formula \(\sqrt{a} \)
Exponentiation formula \((a^m)^n = a^{m \times n}\)

Theorems

Properties of Exponents

Suitable Grade Level

Grades 9-12