Simplify square root of 2x square root of 3y

Solution:

Step 1:

Consider the expression $\sqrt{2x} \cdot \sqrt{3y}$.

Step 2:

Use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$.

Step 3:

Apply the property to the given expression: $\sqrt{2x} \cdot \sqrt{3y} = \sqrt{2x \cdot 3y}$.

Step 4:

Simplify the expression under the square root: $2x \cdot 3y = 6xy$.

Step 5:

Write the simplified expression: $\sqrt{6xy}$.

Knowledge Notes:

When simplifying square roots, there are several key points to remember:

1. Multiplication of Square Roots: The square root of a product is equal to the product of the square roots of each factor. This is expressed as $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

2. Simplifying Inside the Square Root: Before applying the square root, simplify the expression inside as much as possible. This often involves combining like terms or multiplying factors together.

3. Rationalizing the Denominator: If you have a square root in the denominator of a fraction, you typically want to rationalize it by multiplying the numerator and denominator by a suitable square root to eliminate the square root from the denominator.

4. Square Root of Variables: When dealing with variables under a square root, ensure that the variables represent non-negative values, as square roots of negative numbers are not real numbers (unless dealing with complex numbers).

5. Square Root Properties: Familiarize yourself with the properties of square roots, such as the distributive property over multiplication and division, but not over addition and subtraction.

In the given problem, we use the multiplication property of square roots to combine $\sqrt{2x}$ and $\sqrt{3y}$ into a single square root, resulting in the simplified form $\sqrt{6xy}$.