Giải bài 8.27 trang 59

Đề bài

Giá trị của biểu thức ${\left(\sqrt{5}+1\right)}^5-{\left(\sqrt{5}-1\right)}^5$ bằng

 A. 252

 B. 352

 C. 452

 D. 425

Lời giải chi tiết

 Ta có: ${\left(\sqrt{5}+1\right)}^5-{\left(\sqrt{5}-1\right)}^5$

$=\left[{\left(\sqrt{5}\right)}^5+5{\left(\sqrt{5}\right)}^4+10{\left(\sqrt{5}\right)}^3+10{\left(\sqrt{5}\right)}^2+5.\sqrt{5}+1^5\right]$

$-[{\left(\sqrt{5}\right)}^5+5{\left(\sqrt{5}\right)}^4.(-1)+10{\left(\sqrt{5}\right)}^3{(-1)}^2$

$+10{\left(\sqrt{5}\right)}^2(-1{)}^3+5.\sqrt{5}.{(-1)}^4+{(-1)}^5]$

$=\left[{\left(\sqrt{5}\right)}^5+5{\left(\sqrt{5}\right)}^4+10{\left(\sqrt{5}\right)}^3+10{\left(\sqrt{5}\right)}^2+5.\sqrt{5}+1^5\right]$

$-\left[{\left(\sqrt{5}\right)}^5-5{\left(\sqrt{5}\right)}^4+10{\left(\sqrt{5}\right)}^3-10{\left(\sqrt{5}\right)}^2+5.\sqrt{5}-1^5\right]$

$={\left(\sqrt{5}\right)}^5+5{\left(\sqrt{5}\right)}^4+10{\left(\sqrt{5}\right)}^3+10{\left(\sqrt{5}\right)}^2+5.\sqrt{5}+1^5$

$-{\left(\sqrt{5}\right)}^5+5{\left(\sqrt{5}\right)}^4-10{\left(\sqrt{5}\right)}^3+10{\left(\sqrt{5}\right)}^2-5.\sqrt{5}+1^5$

$=10{\left(\sqrt{5}\right)}^4+20{\left(\sqrt{5}\right)}^2+2$

$=10.25+20.5+2=352$.

Chọn B.