Câu 51:

Trục căn thức ở mẫu với giả thiết các biểu thức chữ đều có nghĩa

\(\frac{3}{\sqrt{3}+1} ; \frac{2}{\sqrt{3}-1} ; \frac{2+\sqrt{3}}{2-\sqrt{3}} ; \frac{\mathrm{b}}{3+\sqrt{b}} ; \frac{\mathrm{p}}{2 \sqrt{\mathrm{p}}-1}\)

Lời giải:

+ \(\frac{3}{\sqrt{3}+1}=\frac{3(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}=\frac{3(\sqrt{3}-1)}{3-1}=\frac{3(\sqrt{3}-1)}{2}\)

+ \(\frac{2}{\sqrt{3}-1}=\frac{2(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}=\frac{2(\sqrt{3}+1)}{3-1}=\sqrt{3}+1\)

+ \(\frac{2+\sqrt{3}}{2-\sqrt{3}}=\frac{(2+\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}=\frac{4+4 \sqrt{3}+3}{2^{2}-(\sqrt{3})^{2}}=7+4 \sqrt{3}\)

+ \(\frac{\mathrm{b}}{3+\sqrt{b}}=\frac{\mathrm{b}(3-\sqrt{\mathrm{b}})}{(3+\sqrt{\mathrm{b}})(3-\sqrt{\mathrm{b}})}=\frac{\mathrm{b}(3-\sqrt{\mathrm{b}})}{3^{2}-(\sqrt{\mathrm{b}})^{2}}=\frac{\mathrm{b}(3-\sqrt{\mathrm{b}})}{9-\mathrm{b}}\)

+ \(\frac{p}{2 \sqrt{p}-1}=\frac{p(2 \sqrt{p}+1)}{(2 \sqrt{p}-1)(2 \sqrt{p}+1)}=\frac{p(2 \sqrt{p}+1)}{(2 \sqrt{p})^{2}-1^{2}}=\frac{2 p \sqrt{p}+p}{4 p-1}\)