Which choice is equivalent to the product below?

[tex]\[ \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8} \][/tex]

A. [tex]\( 16 \sqrt{3} \)[/tex]
B. [tex]\( 4 \sqrt{3} \)[/tex]
C. [tex]\( 4 \sqrt{12} \)[/tex]
D. [tex]\( 8 \sqrt{12} \)[/tex]



Answer :

To determine which choice is equivalent to the product [tex]\(\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8}\)[/tex], let's begin by understanding how to handle the multiplication of square roots.

First, recall the property of square roots that states:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]

Using this property, we can combine the square roots one step at a time:

[tex]\[ \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8} = \sqrt{2 \cdot 3 \cdot 8} \][/tex]

Now let's multiply the numbers under the square root:

[tex]\[ 2 \cdot 3 = 6 \][/tex]
[tex]\[ 6 \cdot 8 = 48 \][/tex]

So, we have:

[tex]\[ \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8} = \sqrt{48} \][/tex]

Next, we simplify [tex]\(\sqrt{48}\)[/tex]. To do this, we find the prime factorization of 48:

[tex]\[ 48 = 2^4 \cdot 3 \][/tex]

We can break down [tex]\(\sqrt{48}\)[/tex] using the property that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:

[tex]\[ \sqrt{48} = \sqrt{2^4 \cdot 3} = \sqrt{2^4} \cdot \sqrt{3} \][/tex]

Since [tex]\(\sqrt{2^4} = 2^2 = 4\)[/tex], we get:

[tex]\[ \sqrt{48} = 4 \cdot \sqrt{3} \][/tex]

Thus, the product [tex]\(\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{8}\)[/tex] simplifies to [tex]\(4 \sqrt{3}\)[/tex].

Therefore, the correct choice is:

[tex]\[ \boxed{4 \sqrt{3}} \][/tex]