Multiply square root of 9* square root of 3
The question is asking you to perform a multiplication involving two square roots. Specifically, you are to multiply the square root of 9 by the square root of 3. This involves knowing how to handle square roots and understanding the rules for multiplying them together.
$\sqrt{9} \cdot \sqrt{3}$
Answer
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:
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 $\sqrt{x}$.
Properties of Square Roots: The product of square roots can be combined into a single square root, such that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$.
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).
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.
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.
