To find the equivalent expression to [tex]\(\sqrt{14} \cdot \sqrt{8}\)[/tex], let’s break it down step by step following mathematical principles.
1. Combine the square roots:
We start with the product of the square roots:
[tex]\[
\sqrt{14} \cdot \sqrt{8}
\][/tex]
2. Use the property of square roots:
The property of square roots tells us that the product of two square roots is equal to the square root of the product:
[tex]\[
\sqrt{14} \cdot \sqrt{8} = \sqrt{14 \cdot 8}
\][/tex]
3. Multiply the numbers inside the square root:
Now, we calculate the product of the numbers inside the square root:
[tex]\[
14 \cdot 8 = 112
\][/tex]
So, we have:
[tex]\[
\sqrt{14} \cdot \sqrt{8} = \sqrt{112}
\][/tex]
4. Simplify the square root:
To simplify [tex]\(\sqrt{112}\)[/tex], we can factorize 112 into its prime factors:
[tex]\[
112 = 16 \cdot 7
\][/tex]
Therefore:
[tex]\[
\sqrt{112} = \sqrt{16 \cdot 7}
\][/tex]
5. Use the property of square roots again:
The square root of a product can be expressed as the product of the square roots:
[tex]\[
\sqrt{16 \cdot 7} = \sqrt{16} \cdot \sqrt{7}
\][/tex]
Knowing that [tex]\(\sqrt{16} = 4\)[/tex], we get:
[tex]\[
\sqrt{112} = 4 \cdot \sqrt{7}
\][/tex]
Hence, the product [tex]\(\sqrt{14} \cdot \sqrt{8}\)[/tex] simplifies to [tex]\(4 \sqrt{7}\)[/tex].
Therefore, the equivalent choice is:
A. [tex]\(4 \sqrt{7}\)[/tex].
