Simplify the following radical expression.
$(2+5 \sqrt{2})(6+4 \sqrt{2})$
We need to simplify the expression \((2+5\sqrt{2})(6+4\sqrt{2})\). This expression is a product of two binomials, each containing a radical term. To simplify this, we can apply the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last). We aim to express the result in the form \(A + B\sqrt{C}\), where \(A, B, C\) are integers. The radical part involves \(\sqrt{2}\), and we recall that \(\sqrt{n} \cdot \sqrt{n} = n\) and \(a\sqrt{n} \cdot b\sqrt{m} = ab\sqrt{nm}\).
Using the distributive property:
\((2+5\sqrt{2})(6+4\sqrt{2}) = (2)(6) + (2)(4\sqrt{2}) + (5\sqrt{2})(6) + (5\sqrt{2})(4\sqrt{2})\)
Now, we calculate each of the four products separately:
Now, we combine the calculated terms:
\((12 + 40) + (8\sqrt{2} + 30\sqrt{2})\)
Adding the rational constants: \(12 + 40 = 52\).
Adding the coefficients of the radical terms: \(8\sqrt{2} + 30\sqrt{2} = (8+30)\sqrt{2} = 38\sqrt{2}\).
Combining these results, we get:
\(52 + 38\sqrt{2}\).
The expression \((2+5\sqrt{2})(6+4\sqrt{2})\) simplifies to \(52 + 38\sqrt{2}\). Comparing this with the required form \(A + B\sqrt{C}\), we have \(A = 52\), \(B = 38\), and \(C = 2\). Thus, the simplified expression is \(\boxed{52 + 38\sqrt{2}}\).
In architecture and engineering, radical expressions often appear when calculating diagonal lengths of rectangular structures or areas of shapes involving non-rational proportions. For instance, if you are designing a floor plan where the dimensions are related to the square root of 2, multiplying these dimensions requires simplifying radical expressions to ensure accurate material estimates and precise construction layouts. Mastery of these algebraic manipulations allows you to maintain exact values before converting them to practical measurements.