Multiply square root of 9* square root of 3

Solution:

Step 1:

Express the number 9 as a power of 3, which is $3^2$. Then, represent the multiplication of square roots as $\sqrt{3^2}\cdot \sqrt{3}$.

Step 2:

Extract the square root of the perfect square, which is 3, from under the radical sign, yielding $3\cdot \sqrt{3}$.

Step 3:

Present the final answer in its various acceptable formats:

• Exact Form: $3\sqrt{3}$
• Decimal Form: Approximately $5.19615242$

Knowledge Notes:

To solve the given problem, we need to understand several key concepts:

1. Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of $x$ is written as $\sqrt{x}$.

2. Properties of Square Roots: The product of square roots can be combined into a single square root, such that $\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$.

3. Simplifying Square Roots: When the argument of the square root is a perfect square (like $3^2$), it can be simplified by taking the square root of the perfect square, which results in the base of the power (in this case, 3).

4. Multiplication of Radicals: When multiplying radicals, if the indices (the small number outside the radical sign) are the same, the radicands (the numbers inside the radical sign) can be multiplied together under a single radical.

5. Decimal Approximation: The exact form of a square root may be irrational (cannot be expressed as a simple fraction), so it is often approximated as a decimal for practical use.

In this problem, we used these concepts to rewrite the square root of 9 as the square root of $3^2$, simplified it to 3, and then multiplied it by the square root of 3 to get the final result.