To solve the expression [tex]\((5 + \sqrt{7})^3 (5 - \sqrt{7})^3\)[/tex], we will follow a systematic approach to understand each step.
### Step 1: Define the expressions
Firstly, let's denote:
[tex]\[ a = 5 + \sqrt{7} \][/tex]
[tex]\[ b = 5 - \sqrt{7} \][/tex]
So, our expression can be rewritten as:
[tex]\[ a^3 \cdot b^3 \][/tex]
### Step 2: Use the property of exponents
We know that [tex]\((a \cdot b)^3 = a^3 \cdot b^3 \)[/tex]. Therefore, we can combine [tex]\(a\)[/tex] and [tex]\(b\)[/tex] before cubing:
[tex]\[ a^3 \cdot b^3 = (a \cdot b)^3 \][/tex]
### Step 3: Calculate [tex]\(a \cdot b\)[/tex]
Let's multiply [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
(a \cdot b) = (5 + \sqrt{7})(5 - \sqrt{7})
\][/tex]
We apply the difference of squares formula:
[tex]\[
(a \cdot b) = 5^2 - (\sqrt{7})^2
\][/tex]
Calculate each part:
[tex]\[
5^2 = 25
\][/tex]
[tex]\[
(\sqrt{7})^2 = 7
\][/tex]
Thus:
[tex]\[
a \cdot b = 25 - 7 = 18
\][/tex]
### Step 4: Cube the product
Now we need to cube 18:
[tex]\[
(a \cdot b)^3 = 18^3
\][/tex]
### Step 5: Find the numerical value of [tex]\(18^3\)[/tex]
Using the cubic property:
[tex]\[
18^3 = 5832
\][/tex]
### Conclusion
Thus, the value of the expression [tex]\((5 + \sqrt{7})^3 (5 - \sqrt{7})^3\)[/tex] is:
[tex]\[
\boxed{5832}
\][/tex]
For completeness, the values of each part [tex]\(a^3\)[/tex] and [tex]\(b^3\)[/tex] as previously computed are approximately:
[tex]\(a^3 \approx 446.95160750729644\)[/tex] and [tex]\(b^3 \approx 13.04839249270357\)[/tex].
And their product, as we calculated, is 5832, confirming our final result:
[tex]\[
(5 + \sqrt{7})^3 (5 - \sqrt{7})^3 = 5832
\][/tex]
[tex]\[
\boxed{5832}
\][/tex]
