Question

simplify.
$8\sqrt{8}$

Explanation:

Step1: Simplify the square root

First, we can rewrite \( \sqrt{8} \) as \( \sqrt{4\times2} \). Since \( \sqrt{4\times2}=\sqrt{4}\times\sqrt{2} \) (by the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) for \( a\geq0,b\geq0 \)), and \( \sqrt{4} = 2 \), so \( \sqrt{8}=2\sqrt{2} \).

Step2: Multiply with the coefficient

Now we have \( 8\sqrt{8}=8\times2\sqrt{2} \). Calculating \( 8\times2 = 16 \), so \( 8\sqrt{8}=16\sqrt{2} \). But if we want to express it in the form with a single square root, we can square the coefficient and multiply with the radicand inside the square root. Let's see, \( 16\sqrt{2}=\sqrt{16^{2}\times2}=\sqrt{256\times2}=\sqrt{512} \). Wait, maybe there is a miscalculation here. Wait, actually, let's do it properly.

Wait, the original expression is \( 8\sqrt{8} \). Let's factor 8 inside the square root. \( 8\sqrt{8}=\sqrt{8^{2}\times8} \) (because \( a\sqrt{b}=\sqrt{a^{2}b} \) for \( a\geq0 \)). So \( 8^{2}=64 \), then \( 64\times8 = 512 \). So \( 8\sqrt{8}=\sqrt{512} \). Wait, but maybe the problem is expecting to simplify \( 8\sqrt{8} \) into the form \( \sqrt{x} \), so we can use the formula \( a\sqrt{b}=\sqrt{a^{2}b} \). So \( a = 8 \), \( b = 8 \), then \( a^{2}b=64\times8 = 512 \). So \( 8\sqrt{8}=\sqrt{512} \).

Wait, let's check again. \( 8\sqrt{8} \):

First, simplify \( \sqrt{8} = \sqrt{4\times2}=2\sqrt{2} \). Then \( 8\times2\sqrt{2}=16\sqrt{2} \). Now, to write \( 16\sqrt{2} \) as \( \sqrt{x} \), we square 16 and multiply by 2: \( 16^{2}\times2 = 256\times2 = 512 \). So \( 16\sqrt{2}=\sqrt{512} \). So the simplified form with a single square root is \( \sqrt{512} \).

Answer:

\( \sqrt{512} \) (or if we simplify \( \sqrt{512} \) further, \( \sqrt{256\times2}=16\sqrt{2} \), but the problem seems to want it in the form \( \sqrt{\square} \), so the number inside the square root is 512)