Question

select the correct answer. what is this expression in simplified form? \\((8\sqrt{10})(8\sqrt{5})\\) \\(\circ\\) a. \\(64\sqrt{50}\\) \\(\circ\\) b. \\(16\sqrt{50}\\) \\(\circ\\) c. \\(80\sqrt{2}\\) \\(\circ\\) d. \\(320\sqrt{2}\\)

Explanation:

Step1: Multiply the coefficients and the radicals separately

First, multiply the coefficients (the numbers outside the square roots) and then multiply the radicals (the square root parts) separately. The expression is \((8\sqrt{10})(8\sqrt{5})\). So we have \(8\times8\) for the coefficients and \(\sqrt{10}\times\sqrt{5}\) for the radicals.
\(8\times8 = 64\) and \(\sqrt{10}\times\sqrt{5}=\sqrt{10\times5}=\sqrt{50}\)
So now the expression becomes \(64\sqrt{50}\), but we can simplify \(\sqrt{50}\) further.

Step2: Simplify \(\sqrt{50}\)

We know that \(50 = 25\times2\), so \(\sqrt{50}=\sqrt{25\times2}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a = 25\) and \(b = 2\)), we get \(\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}\)

Step3: Substitute back and simplify the entire expression

Now substitute \(\sqrt{50}=5\sqrt{2}\) into \(64\sqrt{50}\). So we have \(64\times5\sqrt{2}\)
\(64\times5 = 320\), so the simplified form is \(320\sqrt{2}\)

Wait, no, wait. Wait, let's check again. Wait, maybe I made a mistake. Wait, let's redo step 1 and step 2.

Wait, the original expression is \((8\sqrt{10})(8\sqrt{5})\). Let's multiply the coefficients: \(8\times8 = 64\). Multiply the radicals: \(\sqrt{10}\times\sqrt{5}=\sqrt{50}\). So we have \(64\sqrt{50}\). But \(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\). So then \(64\times5\sqrt{2}=320\sqrt{2}\)? Wait, but let's check the options. Option D is \(320\sqrt{2}\). But wait, let's check again. Wait, maybe I messed up the multiplication. Wait, \(8\times8 = 64\), \(\sqrt{10}\times\sqrt{5}=\sqrt{50}\), then \(\sqrt{50}=5\sqrt{2}\), so \(64\times5\sqrt{2}=320\sqrt{2}\). So the answer should be D? Wait, but let's check the options again. Wait, option D is \(320\sqrt{2}\). But let's check the calculation again. Wait, maybe I made a mistake. Wait, let's do it step by step.

First, multiply the two terms: \((8\sqrt{10})(8\sqrt{5})\). Using the property \((a\sqrt{b})(c\sqrt{d})=ac\sqrt{bd}\). So \(a = 8\), \(c = 8\), \(b = 10\), \(d = 5\). So \(ac = 8\times8 = 64\), \(bd = 10\times5 = 50\). So we get \(64\sqrt{50}\). Now simplify \(\sqrt{50}\): \(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\). So then \(64\times5\sqrt{2}=320\sqrt{2}\). So the answer is D. Wait, but let's check the options. Option D is \(320\sqrt{2}\). So that's correct.

Wait, but let's check the options again. The options are:

A. \(64\sqrt{50}\) – but \(\sqrt{50}\) can be simplified, so this is not the simplified form.

B. \(16\sqrt{50}\) – coefficients multiplied wrong (8×8 is 64, not 16)

C. \(80\sqrt{2}\) – 64×5 is 320, not 80

D. \(320\sqrt{2}\) – correct.

So the correct answer is D.

Answer:

D. \(320\sqrt{2}\)