Sure! Let's multiply and simplify the given expressions [tex]\((3 \sqrt{7} - 4)(3 \sqrt{7} + 4)\)[/tex] step-by-step.
We start by recognizing that this looks like the difference of squares formula. The difference of squares formula states that:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
In our case, we can identify:
[tex]\[
a = 3 \sqrt{7} \quad \text{and} \quad b = 4
\][/tex]
Applying the difference of squares formula gives us:
[tex]\[
(3 \sqrt{7} - 4)(3 \sqrt{7} + 4) = (3 \sqrt{7})^2 - 4^2
\][/tex]
Now, we need to calculate each part:
1. Calculate [tex]\((3 \sqrt{7})^2\)[/tex]:
[tex]\[
(3 \sqrt{7})^2 = (3 \sqrt{7}) \times (3 \sqrt{7}) = 3^2 \times (\sqrt{7})^2 = 9 \times 7 = 63
\][/tex]
2. Calculate [tex]\(4^2\)[/tex]:
[tex]\[
4^2 = 4 \times 4 = 16
\][/tex]
Subtract the second result from the first:
[tex]\[
63 - 16 = 47
\][/tex]
Therefore, the simplified result of multiplying [tex]\((3 \sqrt{7} - 4)(3 \sqrt{7} + 4)\)[/tex] is [tex]\(\boxed{47}\)[/tex].
