To simplify the expression [tex]\((\sqrt{2} + \sqrt{3})(\sqrt{5} - \sqrt{7})\)[/tex], we need to apply the distributive property, also known as the FOIL method (First, Outer, Inner, Last). This property allows us to expand the product of two binomials.
The expression we need to simplify is:
[tex]\[
(\sqrt{2} + \sqrt{3})(\sqrt{5} - \sqrt{7})
\][/tex]
We will distribute each term in the first binomial to each term in the second binomial.
1. First: Multiply the first terms in each binomial:
[tex]\[
\sqrt{2} \cdot \sqrt{5} = \sqrt{10}
\][/tex]
2. Outer: Multiply the outer terms in the product:
[tex]\[
\sqrt{2} \cdot (-\sqrt{7}) = -\sqrt{14}
\][/tex]
3. Inner: Multiply the inner terms in the product:
[tex]\[
\sqrt{3} \cdot \sqrt{5} = \sqrt{15}
\][/tex]
4. Last: Multiply the last terms in each binomial:
[tex]\[
\sqrt{3} \cdot (-\sqrt{7}) = -\sqrt{21}
\][/tex]
Now, combine all these products together:
[tex]\[
\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}
\][/tex]
This combination gives the final simplified form of the expression.
So, the simplified expression is:
[tex]\[
\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}
\][/tex]
Numerically, the individual terms are approximately:
- [tex]\(\sqrt{10} \approx 3.1622776601683795\)[/tex]
- [tex]\(-\sqrt{14} \approx -3.7416573867739418\)[/tex]
- [tex]\(\sqrt{15} \approx 3.872983346207417\)[/tex]
- [tex]\(-\sqrt{21} \approx -4.58257569495584\)[/tex]
Adding these values together:
[tex]\[
3.1622776601683795 - 3.7416573867739418 + 3.872983346207417 - 4.58257569495584 \approx -1.288972075353985
\][/tex]
Thus, the final numerical result of the simplified expression is approximately [tex]\(-1.288972075353985\)[/tex].
