Réitigh sqrt{24}(sqrt{2/3}-3sqrt{5/6})

$$\sqrt { 24 } ( \sqrt { \frac { 2 } { 3 } } - 3 \sqrt { \frac { 5 } { 6 } } )$$

Luacháil

$$4-6\sqrt{5}\approx -9.416407865$$

Céimeanna gearra réitigh

Fachtóirigh 24=2^{2}\times 6. Athscríobh fréamh cearnach an toraidh \sqrt{2^{2}\times 6} mar thoradh na bhfréamhacha cearnacha \sqrt{2^{2}}\sqrt{6}. Tóg fréamh chearnach 2^{2}.

$$2\sqrt{6}\left(\sqrt{\frac{2}{3}}-3\sqrt{\frac{5}{6}}\right)$$

Athscríobh fréamh cearnach na roinnte \sqrt{\frac{2}{3}} mar roinnt na bhfréamhacha cearnacha \frac{\sqrt{2}}{\sqrt{3}}.

$$2\sqrt{6}\left(\frac{\sqrt{2}}{\sqrt{3}}-3\sqrt{\frac{5}{6}}\right)$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{3} chun ainmneoir \frac{\sqrt{2}}{\sqrt{3}} a thiontú in uimhir chóimheasta.

$$2\sqrt{6}\left(\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-3\sqrt{\frac{5}{6}}\right)$$

Is é 3 uimhir chearnach \sqrt{3}.

$$2\sqrt{6}\left(\frac{\sqrt{2}\sqrt{3}}{3}-3\sqrt{\frac{5}{6}}\right)$$

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{2} agus \sqrt{3} a iolrú.

$$2\sqrt{6}\left(\frac{\sqrt{6}}{3}-3\sqrt{\frac{5}{6}}\right)$$

Athscríobh fréamh cearnach na roinnte \sqrt{\frac{5}{6}} mar roinnt na bhfréamhacha cearnacha \frac{\sqrt{5}}{\sqrt{6}}.

$$2\sqrt{6}\left(\frac{\sqrt{6}}{3}-3\times \frac{\sqrt{5}}{\sqrt{6}}\right)$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{6} chun ainmneoir \frac{\sqrt{5}}{\sqrt{6}} a thiontú in uimhir chóimheasta.

$$2\sqrt{6}\left(\frac{\sqrt{6}}{3}-3\times \frac{\sqrt{5}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\right)$$

Is é 6 uimhir chearnach \sqrt{6}.

$$2\sqrt{6}\left(\frac{\sqrt{6}}{3}-3\times \frac{\sqrt{5}\sqrt{6}}{6}\right)$$

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{5} agus \sqrt{6} a iolrú.

$$2\sqrt{6}\left(\frac{\sqrt{6}}{3}-3\times \frac{\sqrt{30}}{6}\right)$$

Cealaigh an comhfhachtóir 6 is mó in 3 agus 6.

$$2\sqrt{6}\left(\frac{\sqrt{6}}{3}+\frac{\sqrt{30}}{-2}\right)$$

Úsáid an t-airí dáileach chun 2\sqrt{6} a mhéadú faoi \frac{\sqrt{6}}{3}+\frac{\sqrt{30}}{-2}.

$$2\sqrt{6}\times \frac{\sqrt{6}}{3}+2\sqrt{6}\times \frac{\sqrt{30}}{-2}$$

Scríobh 2\times \frac{\sqrt{6}}{3} mar chodán aonair.

$$\frac{2\sqrt{6}}{3}\sqrt{6}+2\sqrt{6}\times \frac{\sqrt{30}}{-2}$$

Scríobh 2\times \frac{\sqrt{30}}{-2} mar chodán aonair.

$$\frac{2\sqrt{6}}{3}\sqrt{6}+\frac{2\sqrt{30}}{-2}\sqrt{6}$$

Cealaigh -2 agus -2.

$$\frac{2\sqrt{6}}{3}\sqrt{6}-\sqrt{30}\sqrt{6}$$

Scríobh \frac{2\sqrt{6}}{3}\sqrt{6} mar chodán aonair.

$$\frac{2\sqrt{6}\sqrt{6}}{3}-\sqrt{30}\sqrt{6}$$

Fachtóirigh 30=6\times 5. Athscríobh fréamh cearnach an toraidh \sqrt{6\times 5} mar thoradh na bhfréamhacha cearnacha \sqrt{6}\sqrt{5}.

$$\frac{2\sqrt{6}\sqrt{6}}{3}-\sqrt{6}\sqrt{5}\sqrt{6}$$

Méadaigh \sqrt{6} agus \sqrt{6} chun 6 a fháil.

$$\frac{2\sqrt{6}\sqrt{6}}{3}-6\sqrt{5}$$

Chun cothromóidí a shuimiú nó a dhealú, fairsingigh iad chun a n-ainmneoirí a mheaitseáil. Méadaigh -6\sqrt{5} faoi \frac{3}{3}.

$$\frac{2\sqrt{6}\sqrt{6}}{3}+\frac{3\left(-6\right)\sqrt{5}}{3}$$

Tá an t-ainmneoir céanna ag \frac{2\sqrt{6}\sqrt{6}}{3} agus \frac{3\left(-6\right)\sqrt{5}}{3} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

$$\frac{2\sqrt{6}\sqrt{6}+3\left(-6\right)\sqrt{5}}{3}$$

Déan iolrúcháin in 2\sqrt{6}\sqrt{6}+3\left(-6\right)\sqrt{5}.

$$\frac{12-18\sqrt{5}}{3}$$