Simplify ( square root of 360mn)/(3 square root of 5)

Solution:

Step 1:

Merge $\sqrt{360mn}$ with $\sqrt{5}$ under a single radical to get $\frac{\sqrt{\frac{360mn}{5}}}{3}$.

Step 2:

Identify and eliminate common factors between $360$ and $5$.

Step 2.1:

Extract $5$ from $360mn$ to obtain $\frac{\sqrt{\frac{5(72mn)}{5}}}{3}$.

Step 2.2:

Proceed to simplify by removing common factors.

Step 2.2.1:

Isolate $5$ from the denominator as well, resulting in $\frac{\sqrt{\frac{5(72mn)}{5(1)}}}{3}$.

Step 2.2.2:

Eliminate the $5$ from both numerator and denominator to get $\frac{\sqrt{\frac{\cancel{5}(72mn)}{\cancel{5} \cdot 1}}}{3}$.

Step 2.2.3:

Express the simplified fraction as $\frac{\sqrt{\frac{72mn}{1}}}{3}$.

Step 2.2.4:

Simplify further by dividing $72mn$ by $1$, yielding $\frac{\sqrt{72mn}}{3}$.

Step 3:

Simplify the square root in the numerator.

Step 3.1:

Express $72mn$ as the product of a perfect square and other factors.

Step 3.1.1:

Factor out $36$ from $72$, giving $\frac{\sqrt{36(2mn)}}{3}$.

Step 3.1.2:

Represent $36$ as $6^2$, leading to $\frac{\sqrt{6^2 \cdot 2mn}}{3}$.

Step 3.1.3:

Introduce parentheses for clarity, resulting in $\frac{\sqrt{6^2 \cdot (2mn)}}{3}$.

Step 3.1.4:

Maintain the parentheses, which gives $\frac{\sqrt{6^2 \cdot (2mn)}}{3}$.

Step 3.2:

Extract terms from under the radical, simplifying to $\frac{6\sqrt{2mn}}{3}$.

Step 4:

Reduce the fraction by removing common factors between $6$ and $3$.

Step 4.1:

Factor out $3$ from the numerator to get $\frac{3(2\sqrt{2mn})}{3}$.

Step 4.2:

Proceed to cancel out common factors.

Step 4.2.1:

Factor $3$ from the denominator as well, resulting in $\frac{3(2\sqrt{2mn})}{3(1)}$.

Step 4.2.2:

Eliminate the common $3$, simplifying to $\frac{\cancel{3}(2\sqrt{2mn})}{\cancel{3} \cdot 1}$.

Step 4.2.3:

Express the simplified expression as $\frac{2\sqrt{2mn}}{1}$.

Step 4.2.4:

Finally, divide $2\sqrt{2mn}$ by $1$ to obtain the simplified result $2\sqrt{2mn}$.

Knowledge Notes:

The problem involves simplifying a radical expression, which is a common task in algebra. Here are the relevant knowledge points:

1. Radical Simplification: The process of simplifying square roots (or other roots) to their simplest form. This often involves factoring out perfect squares and reducing the expression to a simpler radical form.

2. Combining Radicals: When radicals (square roots) have the same index, they can be combined under a single radical sign by multiplying the values inside the radical.

3. Rationalizing the Denominator: This process involves eliminating radicals from the denominator of a fraction. In this problem, we simplify by combining radicals and then canceling common factors.

4. Factoring Perfect Squares: Recognizing perfect squares within a radical allows us to take the square root more easily. For example, $36$ is a perfect square, and its square root is $6$.

5. Simplifying Fractions: When a fraction has common factors in the numerator and denominator, we can cancel them out to simplify the fraction.

6. LaTeX Formatting: The solution uses LaTeX to format mathematical expressions, which is a typesetting system widely used for mathematical and scientific documents. LaTeX allows for clear and precise rendering of complex formulas.

In solving the given problem, we used these concepts to systematically simplify the expression step by step, ensuring that each action is mathematically valid and leads to a simpler form.