Question

simplify. \\(sqrt{128}\\) suggested tutorial: learn it: simplify the square root of a real number. resources read it ask your teacher practice another tgelemalg5 8.2.028.

Explanation:

Step1: Factor 128 into perfect square and other

We know that \(128 = 64\times2\), and \(64\) is a perfect square (\(8^2\) or \(16^2\)? Wait, \(8^2 = 64\), \(16^2=256\). Wait, \(64\times2 = 128\), and \(64 = 8^2\), but also \(128= 64\times2=8^2\times2\), but wait, the suggested answer was \(16\sqrt{2}\), so maybe I made a mistake. Wait, \(128 = 64\times2=8^2\times2\), but \(64\times2 = 128\), but \(128 = 16\times8=16\times4\times2 = 64\times2\)? Wait, no, \(16\times8 = 128\), and \(8 = 4\times2\), so \(16\times4\times2=64\times2\). Wait, \(128 = 64\times2\), and \(\sqrt{64\times2}=\sqrt{64}\times\sqrt{2}=8\sqrt{2}\)? But the suggested answer was \(16\sqrt{2}\), that means I factored wrong. Wait, maybe \(128 = 256\div2\)? No, wait, \(16^2 = 256\), no. Wait, \(128 = 16\times8\), and \(8 = 4\times2\), so \(16\times4\times2 = 64\times2\), so \(\sqrt{128}=\sqrt{64\times2}=\sqrt{64}\times\sqrt{2}=8\sqrt{2}\)? But the suggested answer in the image was \(16\sqrt{2}\), that must be a mistake? Wait, no, maybe I miscalculated. Wait, \(16\sqrt{2}\) squared is \(16^2\times2 = 256\times2 = 512\), which is not 128. Wait, that's wrong. Wait, \(8\sqrt{2}\) squared is \(64\times2 = 128\), so the correct simplification is \(8\sqrt{2}\), but the image has a cross on \(16\sqrt{2}\), meaning that's wrong. Wait, the problem is to simplify \(\sqrt{128}\). Let's do it correctly.

Step1: Factor 128 into perfect square factor

We need to express \(128\) as a product of a perfect square and another integer. Let's find the largest perfect square that divides \(128\). The perfect squares are \(1,4,9,16,25,36,49,64,81,100,121,144,\dots\). The largest one that divides \(128\) is \(64\) (since \(128\div64 = 2\)). So, \(128 = 64\times2\).

Step2: Apply square root property

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a\geq0\), \(b\geq0\), we have:
\(\sqrt{128}=\sqrt{64\times2}=\sqrt{64}\times\sqrt{2}\)

Step3: Simplify \(\sqrt{64}\)

Since \(8^2 = 64\), \(\sqrt{64}=8\). So, \(\sqrt{128}=8\sqrt{2}\).

Wait, but the image had \(16\sqrt{2}\) marked wrong, so the correct simplification is \(8\sqrt{2}\). But maybe the original problem was \(\sqrt{512}\)? Because \(16\sqrt{2}\) squared is \(256\times2 = 512\). Maybe a typo. But assuming the problem is \(\sqrt{128}\), the correct simplification is \(8\sqrt{2}\). But let's check again.

Wait, \(128 = 2^7\), because \(2^7 = 128\). So, \(\sqrt{2^7}=\sqrt{2^6\times2}=2^3\sqrt{2}=8\sqrt{2}\), since \(2^6=(2^3)^2 = 8^2\). So that's correct. So the mistake in the suggested answer (the \(16\sqrt{2}\)) is wrong, the correct answer is \(8\sqrt{2}\).

But maybe the problem was \(\sqrt{512}\)? Let's check \(\sqrt{512}\). \(512 = 256\times2 = 16^2\times2\), so \(\sqrt{512}=\sqrt{16^2\times2}=16\sqrt{2}\). Ah, so maybe the problem was \(\sqrt{512}\) instead of \(\sqrt{128}\). So if the problem is \(\sqrt{512}\), then:

Step1: Factor 512 into perfect square and other

\(512 = 256\times2 = 16^2\times2\)

Step2: Apply square root property

\(\sqrt{512}=\sqrt{16^2\times2}=\sqrt{16^2}\times\sqrt{2}=16\sqrt{2}\)

So probably a typo in the problem, and it's \(\sqrt{512}\) instead of \(\sqrt{128}\). So assuming that, the simplification is \(16\sqrt{2}\).

Answer:

\(8\sqrt{2}\) (if problem is \(\sqrt{128}\)) or \(16\sqrt{2}\) (if problem is \(\sqrt{512}\))

But based on the image, the problem is \(\sqrt{128}\), so the correct answer is \(8\sqrt{2}\), but the suggested answer was wrong. However, if we consider the suggested answer, maybe the problem was \(\sqrt{512}\), so the answer is \(16\sqrt{2}\).