Dạng 1: Tính giá trị biểu thức [Rút gọn biểu thức rồi thay số (nếu đc)]
1) Tính giá trị biểu thức B = \(\sqrt{x-1+2\sqrt[3]{x\sqrt{x}+3x+3\sqrt{x}+1}}\), vs x = 5
2) Tính giá trị biểu thức C = \(\sqrt{2x-1+2\sqrt{x^2-x}+\sqrt{2x-1-2\sqrt{x^2-x}}}\), vs x = 4
3) Tính giá trị biểu thức D = \(\frac{\sqrt[3]{x\sqrt{x}\left(3x+1\right)+x^2\left(3+x\right)}}{\sqrt{x}+1}-\sqrt{x}\), vs x = 10
4) Tính giá trị biểu thức E = \(\sqrt{\sqrt[4]{x}+1-2\sqrt[8]{x}+1}\), vs x = 256
5) Cho x = \(\frac{\left(\sqrt{5}+2\right)\sqrt{3\sqrt{5}-6}}{\sqrt{4+\sqrt{9-4\sqrt{5}}}}\), tính giá trị biểu thức A = \(\left(x^4-5x^2+5\right)^{2014}\)
1) \(B=\sqrt{x-1+2\sqrt[3]{x\sqrt{x}+3x+3\sqrt{x}+1}}\)
\(B=\sqrt{x-1+2\sqrt[3]{\sqrt{x^3}+3x+3\sqrt{x}+1}}\)
\(B=\sqrt{x-1+2\sqrt[3]{\left(\sqrt{x}+1\right)^3}}\)
\(B=\sqrt{x-1+2\left(\sqrt{x}+1\right)}\)
\(B=\sqrt{x-1+2\sqrt{x}+2}\)
\(B=\sqrt{\left(\sqrt{x}+1\right)^2}\)
\(B=\sqrt{x}+1\)
\(B=\sqrt{5}+1\)
2) Sửa đề :
\(C=\sqrt{2x-1+2\sqrt{x^2-x}}+\sqrt{2x-1-2\sqrt{x^2-x}}\)
\(C=\sqrt{x+2\sqrt{x\left(x-1\right)}+x-1}+\sqrt{x-2\sqrt{x\left(x-1\right)}+x-1}\)
\(C=\sqrt{\left(\sqrt{x}+\sqrt{x-1}\right)^2}+\sqrt{\left(\sqrt{x}-\sqrt{x-1}\right)^2}\)
\(C=\sqrt{x}+\sqrt{x-1}+\sqrt{x}-\sqrt{x-1}\)
\(C=2\sqrt{x}\)
\(C=2\cdot\sqrt{4}=4\)
5) \(x=\frac{\left(\sqrt{5}+2\right)\sqrt{3\sqrt{5}-6}}{\sqrt{4+\sqrt{9-4\sqrt{5}}}}\)
\(\Leftrightarrow x=\frac{\left(\sqrt{5}+2\right)\cdot\sqrt{3}\cdot\sqrt{\sqrt{5}-2}}{\sqrt{4+\sqrt{\left(\sqrt{5}-2\right)^2}}}\)
\(\Leftrightarrow x=\frac{\left(\sqrt{5}+2\right)\cdot\sqrt{3}\cdot\sqrt{\sqrt{5}-2}}{\sqrt{4+\sqrt{5}-2}}\)
\(\Leftrightarrow x=\frac{\left(\sqrt{5}+2\right)\cdot\sqrt{3}\cdot\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+2}}\)
\(\Leftrightarrow x=\sqrt{3}\cdot\sqrt{\sqrt{5}+2}\cdot\sqrt{\sqrt{5}-2}\)
\(\Leftrightarrow x=\sqrt{3}\cdot\sqrt{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}\)
\(\Leftrightarrow x=\sqrt{3}\cdot\sqrt{5-4}\)
\(\Leftrightarrow x=\sqrt{3}\)
Thay vào A ta được :
\(A=\left[\left(\sqrt{3}\right)^4-5\cdot\left(\sqrt{3}\right)^2+5\right]^{2014}\)
\(A=\left(9-5\cdot3+5\right)^{2014}\)
\(A=\left(-1\right)^{2014}\)
\(A=1\)
Vậy...
3) \(D=\frac{\sqrt[3]{x\sqrt{x}\left(3x+1\right)+x^2\left(3+x\right)}}{\sqrt{x}+1}-\sqrt{x}\)
\(=\frac{\sqrt[3]{x^3+3x^2\sqrt{x}+3x^2+x\sqrt{x}}}{\sqrt{x}+1}-\sqrt{x}\)
\(=\frac{\sqrt[3]{\left(x+\sqrt{x}\right)^3}}{\sqrt{x}+1}-\sqrt{x}\)
\(=\frac{x+\sqrt{x}}{\sqrt{x}+1}-\sqrt{x}\)
\(=0\)