Math Problem Statement

\sqrt{9}\cdot\sqrt[3]{9}

Solution

The expression combines a square root and a cube root of 99. Here's the detailed calculation:

Expression:

993\sqrt{9} \cdot \sqrt[3]{9}

Step 1: Evaluate each root separately.

  1. 9\sqrt{9}: The square root of 99 is 33, because 32=93^2 = 9. 9=3\sqrt{9} = 3

  2. 93\sqrt[3]{9}: The cube root of 99 is the number xx such that x3=9x^3 = 9. This is left in its radical form as: 93\sqrt[3]{9}

Step 2: Multiply the results.

The product is: [ \sqrt{9} \cdot \sqrt[3]{9} = 3 \cdot \sqrt[3]{9} ]

Final Answer:

3933\sqrt[3]{9}

Would you like further explanations or to see this in another form (like decimal approximation)?


Related Questions:

  1. How do square and cube roots differ in their applications?
  2. Can 93\sqrt[3]{9} be simplified further or approximated numerically?
  3. What is the geometric interpretation of square and cube roots?
  4. How would this calculation change if it were 94\sqrt[4]{9}?
  5. What happens if we multiply 9393\sqrt[3]{9} \cdot \sqrt[3]{9}?

Tip:

Square and cube roots can often be combined using the rule of exponents: am/n=amna^{m/n} = \sqrt[n]{a^m}.

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Math Problem Analysis

Mathematical Concepts

Square Roots
Cube Roots
Radicals
Basic Arithmetic Operations

Formulas

Square root: \( \sqrt{a} \) represents the non-negative number whose square is \(a\).
Cube root: \( \sqrt[3]{a} \) represents the number whose cube is \(a\).
Radical multiplication: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) (if both are square roots).

Theorems

Rules of Radicals

Suitable Grade Level

Grades 8-10