Simplify.
$8 \cdot \sqrt{64} + 9 \cdot \sqrt{4}$
The problem asks to simplify the arithmetic expression \(8 \cdot \sqrt{64} + 9 \cdot \sqrt{4}\). This expression involves basic arithmetic operations: multiplication, addition, and the square root function. To simplify it, we need to evaluate the square root terms first, then perform the multiplications, and finally add the results. The square root of a non-negative number \(x\) is a number \(y\) such that \(y^2 = x\) and \(y \ge 0\). Specifically, we are looking for \(\sqrt{64}\) and \(\sqrt{4}\). Since \(8^2 = 64\) and \(2^2 = 4\), we know \(\sqrt{64} = 8\) and \(\sqrt{4} = 2\).
We replace \(\sqrt{64}\) with \(8\) and \(\sqrt{4}\) with \(2\) in the original expression:
\[8 \cdot \sqrt{64} + 9 \cdot \sqrt{4} = 8 \cdot 8 + 9 \cdot 2\]
Following the order of operations, we perform the multiplications next:
\[8 \cdot 8 = 64\]
\[9 \cdot 2 = 18\]
Now we add the results of these multiplications together:
\[64 + 18 = 82\]
Thus, the expression simplifies to \(82\).
Simplifying expressions involving square roots is a foundational skill in algebra, commonly used in areas like geometry for calculating distances or in engineering for scaling values. For example, if you are designing a rectangular garden and need to calculate the total length of fencing required based on multiple segments that involve square roots of area units, simplifying these expressions first makes your calculations much easier and faster. Mastering this allows you to handle more complex physical models with confidence.