Sure, let's simplify the expression [tex]\(\sqrt{6} \cdot \sqrt{8}\)[/tex] step-by-step:
1. Using the property of square roots:
The first step is to use the property of square roots which states that [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]. In our case, we can combine the two square roots into a single square root:
[tex]\[
\sqrt{6} \cdot \sqrt{8} = \sqrt{6 \cdot 8}
\][/tex]
2. Multiply the numbers inside the square root:
Next, we need to multiply the numbers inside the square root:
[tex]\[
6 \cdot 8 = 48
\][/tex]
3. Take the square root of the product:
Now, we take the square root of 48:
[tex]\[
\sqrt{48}
\][/tex]
4. Simplify the square root if possible:
The square root of 48 can be simplified by recognizing that 48 can be factored into 16 and 3 (since 48 = 16 [tex]\(\times\)[/tex] 3). The square root of 16 is 4, so we can further simplify as follows:
[tex]\[
\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}
\][/tex]
After all these steps, we found that:
[tex]\[
\sqrt{6} \cdot \sqrt{8} = 4\sqrt{3}
\][/tex]
For the numeric value, the square root of 48 is approximately:
[tex]\[
\sqrt{48} \approx 6.928203230275509
\][/tex]
Therefore, simplified and numerically verified:
[tex]\[
\sqrt{6} \cdot \sqrt{8} = 4\sqrt{3} \approx 6.928203230275509
\][/tex]
So, the simplified for [tex]\(\sqrt{6} \cdot \sqrt{8}\)[/tex] in its numerical form is approximately 6.928203230275509.
