What is the following product?
$\sqrt{12} \cdot \sqrt{18}$
We are given the expression \(\sqrt{12} \cdot \sqrt{18}\). To solve this, we can use the product property of square roots, which states that for any non-negative real numbers \(x\) and \(y\), the product \(\sqrt{x} \cdot \sqrt{y}\) is equal to \(\sqrt{x \cdot y}\). This property allows us to combine the two radicals into a single radical expression.
Applying the property, we have:
\[\sqrt{12} \cdot \sqrt{18} = \sqrt{12 \cdot 18}\]
Next, calculate the product inside the square root:
\[12 \cdot 18 = 216\]
So, the expression becomes \(\sqrt{216}\).
To simplify \(\sqrt{216}\), we look for the largest perfect square factor of \(216\). We can factorize \(216\) as:
\[216 = 36 \cdot 6\]
Since \(36\) is a perfect square (\(6^2 = 36\)), we can use the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) to simplify the expression:
\[\sqrt{216} = \sqrt{36 \cdot 6} = \sqrt{36} \cdot \sqrt{6} = 6\sqrt{6}\]
Thus, the value of the product is \(6\sqrt{6}\).
Square roots are frequently used in geometry and construction. For instance, if you are designing a square garden with an area of 216 square meters, you might need to find the side length, which involves simplifying \(\sqrt{216}\). Recognizing that \(\sqrt{216} = 6\sqrt{6}\) provides an exact and easy-to-use measurement for practical applications, like calculating fencing requirements.