Réitigh (sqrt{32}+sqrt{05}-2sqrt{1/3})-(sqrt{1/8}-sqrt{75})

$$( \sqrt { 32 } + \sqrt { 05 } - 2 \sqrt { \frac { 1 } { 3 } } ) - ( \sqrt { \frac { 1 } { 8 } } - \sqrt { 75 } )$$

Luacháil

$$\frac{13\sqrt{3}}{3}+\frac{15\sqrt{2}}{4}\approx 12.808854358$$

Céimeanna gearra réitigh

Fachtóirigh 32=4^{2}\times 2. Athscríobh fréamh cearnach an toraidh \sqrt{4^{2}\times 2} mar thoradh na bhfréamhacha cearnacha \sqrt{4^{2}}\sqrt{2}. Tóg fréamh chearnach 4^{2}.

$$4\sqrt{2}+\sqrt{0\times 5}-2\sqrt{\frac{1}{3}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Méadaigh 0 agus 5 chun 0 a fháil.

$$4\sqrt{2}+\sqrt{0}-2\sqrt{\frac{1}{3}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Áirigh fréamh chearnach 0 agus faigh 0.

$$4\sqrt{2}+0-2\sqrt{\frac{1}{3}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Athscríobh fréamh cearnach na roinnte \sqrt{\frac{1}{3}} mar roinnt na bhfréamhacha cearnacha \frac{\sqrt{1}}{\sqrt{3}}.

$$4\sqrt{2}+0-2\times \frac{\sqrt{1}}{\sqrt{3}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Áirigh fréamh chearnach 1 agus faigh 1.

$$4\sqrt{2}+0-2\times \frac{1}{\sqrt{3}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{3} chun ainmneoir \frac{1}{\sqrt{3}} a thiontú in uimhir chóimheasta.

$$4\sqrt{2}+0-2\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Is é 3 uimhir chearnach \sqrt{3}.

$$4\sqrt{2}+0-2\times \frac{\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

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

$$4\sqrt{2}+0+\frac{-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Chun cothromóidí a shuimiú nó a dhealú, fairsingigh iad chun a n-ainmneoirí a mheaitseáil. Méadaigh 4\sqrt{2}+0 faoi \frac{3}{3}.

$$\frac{3\left(4\sqrt{2}+0\right)}{3}+\frac{-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Tá an t-ainmneoir céanna ag \frac{3\left(4\sqrt{2}+0\right)}{3} agus \frac{-2\sqrt{3}}{3} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

$$\frac{3\left(4\sqrt{2}+0\right)-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Déan iolrúcháin in 3\left(4\sqrt{2}+0\right)-2\sqrt{3}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Athscríobh fréamh cearnach na roinnte \sqrt{\frac{1}{8}} mar roinnt na bhfréamhacha cearnacha \frac{\sqrt{1}}{\sqrt{8}}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{1}}{\sqrt{8}}-\sqrt{75}\right)$$

Áirigh fréamh chearnach 1 agus faigh 1.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{1}{\sqrt{8}}-\sqrt{75}\right)$$

Fachtóirigh 8=2^{2}\times 2. Athscríobh fréamh cearnach an toraidh \sqrt{2^{2}\times 2} mar thoradh na bhfréamhacha cearnacha \sqrt{2^{2}}\sqrt{2}. Tóg fréamh chearnach 2^{2}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{1}{2\sqrt{2}}-\sqrt{75}\right)$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{2} chun ainmneoir \frac{1}{2\sqrt{2}} a thiontú in uimhir chóimheasta.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}-\sqrt{75}\right)$$

Is é 2 uimhir chearnach \sqrt{2}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{2\times 2}-\sqrt{75}\right)$$

Méadaigh 2 agus 2 chun 4 a fháil.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{4}-\sqrt{75}\right)$$

Fachtóirigh 75=5^{2}\times 3. Athscríobh fréamh cearnach an toraidh \sqrt{5^{2}\times 3} mar thoradh na bhfréamhacha cearnacha \sqrt{5^{2}}\sqrt{3}. Tóg fréamh chearnach 5^{2}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{4}-5\sqrt{3}\right)$$

Chun cothromóidí a shuimiú nó a dhealú, fairsingigh iad chun a n-ainmneoirí a mheaitseáil. Méadaigh -5\sqrt{3} faoi \frac{4}{4}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{4}+\frac{4\left(-5\right)\sqrt{3}}{4}\right)$$

Tá an t-ainmneoir céanna ag \frac{\sqrt{2}}{4} agus \frac{4\left(-5\right)\sqrt{3}}{4} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\frac{\sqrt{2}+4\left(-5\right)\sqrt{3}}{4}$$

Déan iolrúcháin in \sqrt{2}+4\left(-5\right)\sqrt{3}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\frac{\sqrt{2}-20\sqrt{3}}{4}$$

Chun cothromóidí a shuimiú nó a dhealú, fairsingigh iad chun a n-ainmneoirí a mheaitseáil. Is é an t-iolrach is lú coitianta de 3 agus 4 ná 12. Méadaigh \frac{12\sqrt{2}-2\sqrt{3}}{3} faoi \frac{4}{4}. Méadaigh \frac{\sqrt{2}-20\sqrt{3}}{4} faoi \frac{3}{3}.

$$\frac{4\left(12\sqrt{2}-2\sqrt{3}\right)}{12}-\frac{3\left(\sqrt{2}-20\sqrt{3}\right)}{12}$$

Tá an t-ainmneoir céanna ag \frac{4\left(12\sqrt{2}-2\sqrt{3}\right)}{12} agus \frac{3\left(\sqrt{2}-20\sqrt{3}\right)}{12} agus, mar sin, is féidir iad a dhealú trína n-uimhreoirí a dhealú.

$$\frac{4\left(12\sqrt{2}-2\sqrt{3}\right)-3\left(\sqrt{2}-20\sqrt{3}\right)}{12}$$

Déan iolrúcháin in 4\left(12\sqrt{2}-2\sqrt{3}\right)-3\left(\sqrt{2}-20\sqrt{3}\right).

$$\frac{48\sqrt{2}-8\sqrt{3}-3\sqrt{2}+60\sqrt{3}}{12}$$

Déan áirimh in 48\sqrt{2}-8\sqrt{3}-3\sqrt{2}+60\sqrt{3}.

$$\frac{45\sqrt{2}+52\sqrt{3}}{12}$$