Sure, let's simplify the expression [tex]\((5+\sqrt{3})(7+\sqrt{5})\)[/tex] step-by-step.
To simplify, we apply the distributive property (also known as the FOIL method for binomials):
[tex]\[
(5 + \sqrt{3})(7 + \sqrt{5})
\][/tex]
The FOIL method involves multiplying each term in the first binomial by each term in the second binomial and then adding the results:
1. First: Multiply the first terms in each binomial.
[tex]\[
5 \cdot 7 = 35
\][/tex]
2. Outer: Multiply the outer terms in the binomials.
[tex]\[
5 \cdot \sqrt{5} = 5\sqrt{5} \approx 11.18034
\][/tex]
3. Inner: Multiply the inner terms in the binomials.
[tex]\[
\sqrt{3} \cdot 7 = 7\sqrt{3} \approx 12.12436
\][/tex]
4. Last: Multiply the last terms in each binomial.
[tex]\[
\sqrt{3} \cdot \sqrt{5} = \sqrt{15} \approx 3.87298
\][/tex]
Now, we sum these results to get the simplified form of the expression:
[tex]\[
35 + 5\sqrt{5} + 7\sqrt{3} + \sqrt{15}
\][/tex]
Substituting the approximated values we calculated:
[tex]\[
35 + 11.18034 + 12.12436 + 3.87298 = 62.17768
\][/tex]
So, the simplified expression is:
[tex]\[
(5 + \sqrt{3})(7 + \sqrt{5}) = 35 + 5\sqrt{5} + 7\sqrt{3} + \sqrt{15} \approx 62.17768
\][/tex]
