Answer:
[tex](8 - \sqrt{2}) \cdot (2 + \sqrt{8}) = 12 + 14\; \sqrt{2}[/tex].
Step-by-step explanation:
Step One: Simplify the square roots.
[tex]8 = 4 \times 2 = 2^2 \times 2[/tex].
The square root of 8 [tex]\sqrt{8}[/tex] can be simplified as the product of an integer and the square root of 2:
[tex]\sqrt{8} = \sqrt{4 \times 2} = \sqrt{2^2 \times 2} = \sqrt{2^{2}} \times \sqrt{2} = 2 \; \sqrt{2}[/tex].
As a result,
[tex](8 - \sqrt{2}) \cdot (2 + \sqrt{8}) = (8 - \sqrt{2}) \cdot (2 + 2\; \sqrt{2})[/tex].
Step Two: Expand the product of the two binomials.
[tex](8 - \sqrt{2}) \cdot (2 + 2\; \sqrt{2})[/tex] is the product of two binomials:
- Binomial One: [tex]8 - \sqrt{2}[/tex].
- Binomial Two: [tex]2 + 2\; \sqrt{2}[/tex]
Start by applying the distributive law to the first binomial. Multiply each term in the first binomial (without brackets) with the second binomial (with brackets)
[tex]({\bf 8} - {\bf \sqrt{2}}) \cdot {(2 + 2\; \sqrt{2})}\\= [{\bf 8} \cdot {(2 + 2\; \sqrt{2})}] - [{\bf \sqrt{2}} \cdot {(2 + 2\; \sqrt{2})}][/tex]
Now, apply the distributive law once again to terms in the second binomial.
[tex][8 \cdot ({\bf 2} + {\bf 2\; \sqrt{2}})] - [\sqrt{2} \cdot ({\bf 2} + {\bf 2\; \sqrt{2}})]\\= [8 \times {\bf 2} + 8 \times {\bf 2\;\sqrt{2}}] - [\sqrt{2} \times {\bf 2} + \sqrt{2} \times {\bf 2\; \sqrt{2}}][/tex].
Step Three: Simplify the expression.
The square of a square root is the same as the number under the square root. For example, [tex]\sqrt{2} \times \sqrt{2} = (\sqrt{2})^{2} = 2[/tex].
[tex][8 \times 2 + 8 \times 2\;\sqrt{2}] - [\sqrt{2} \times 2 + 2 \sqrt{2} \times \sqrt{2}]\\ =[16 + 16 \;\sqrt{2}] - [2 \;\sqrt{2} + 4]\\= 16 + 16\;\sqrt{2} - 2\; \sqrt{2} - 4[/tex].
Combine the terms with the square root of two and those without the square root of two:
[tex]16 + 16\;\sqrt{2} - 2\; \sqrt{2} - 4\\= (16 - 4) + (16 \; \sqrt{2} - 2\; \sqrt{2})[/tex].
Factor the square root of two out of the second term:
[tex](16 - 4) + (16 \; \sqrt{2} - 2\; \sqrt{2})\\= (16 - 4) + (16 - 2) \; \sqrt{2} \\= 12 - 14 \; \sqrt{2}[/tex].
Combining the steps:
[tex](8 - \sqrt{2}) \cdot (2 + 2\; \sqrt{2})\\= [8 \cdot (2 + 2\; \sqrt{2})] - [\sqrt{2} \cdot (2 + 2\; \sqrt{2})]\\= [8 \times 2 + 8 \times 2\;\sqrt{2}] - [\sqrt{2} \times 2 + \sqrt{2} \times 2 \;\sqrt{2}]\\= [16 + 16 \;\sqrt{2}] - [2 \;\sqrt{2} + 2 \times (\sqrt{2})^{2}]\\= [16 + 16 \;\sqrt{2}] - [2 \;\sqrt{2} + 2 \times 2]\\= [16 + 16 \;\sqrt{2}] - [2 \;\sqrt{2} + 4]\\= 16 + 16\;\sqrt{2} - 2\; \sqrt{2} - 4\\= (16 - 4) + (16 \; \sqrt{2} - 2\; \sqrt{2})\\[/tex]
[tex]= (16 - 4) + (16 - 2) \; \sqrt{2}\\= 12 - 14 \; \sqrt{2}[/tex].
