Selesaikan frac{sqrt{2}-1}{2sqrt{2}+1}*frac{sqrt{2}+1}{-sqrt{2}+1}=

$$\frac { \sqrt { 2 } - 1 } { 2 \sqrt { 2 } + 1 } \cdot \frac { \sqrt { 2 } + 1 } { - \sqrt { 2 } + 1 } =$$

Evaluasi

$$\frac{-\sqrt{2}-3}{7}\approx -0,630601937$$

Langkah Solusi

Rasionalkan penyebut dari \frac{\sqrt{2}-1}{2\sqrt{2}+1} dengan mengalikan pembilang dan penyebut dengan 2\sqrt{2}-1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{\left(2\sqrt{2}+1\right)\left(2\sqrt{2}-1\right)}\times \frac{\sqrt{2}+1}{-\sqrt{2}+1}$$

Sederhanakan \left(2\sqrt{2}+1\right)\left(2\sqrt{2}-1\right). Perkalian dapat diubah menjadi bentuk selisih dua kuadrat menggunakan aturan: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{\left(2\sqrt{2}\right)^{2}-1^{2}}\times \frac{\sqrt{2}+1}{-\sqrt{2}+1}$$

Luaskan \left(2\sqrt{2}\right)^{2}.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{2^{2}\left(\sqrt{2}\right)^{2}-1^{2}}\times \frac{\sqrt{2}+1}{-\sqrt{2}+1}$$

Hitung 2 sampai pangkat 2 dan dapatkan 4.

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

Kuadrat \sqrt{2} adalah 2.

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

Kalikan 4 dan 2 untuk mendapatkan 8.

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

Hitung 1 sampai pangkat 2 dan dapatkan 1.

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

Kurangi 1 dari 8 untuk mendapatkan 7.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\sqrt{2}+1}{-\sqrt{2}+1}$$

Rasionalkan penyebut dari \frac{\sqrt{2}+1}{-\sqrt{2}+1} dengan mengalikan pembilang dan penyebut dengan -\sqrt{2}-1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{\left(-\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}$$

Sederhanakan \left(-\sqrt{2}+1\right)\left(-\sqrt{2}-1\right). Perkalian dapat diubah menjadi bentuk selisih dua kuadrat menggunakan aturan: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{\left(-\sqrt{2}\right)^{2}-1^{2}}$$

Hitung -\sqrt{2} sampai pangkat 2 dan dapatkan \left(\sqrt{2}\right)^{2}.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Hitung 1 sampai pangkat 2 dan dapatkan 1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{\left(\sqrt{2}\right)^{2}-1}$$

Kuadrat \sqrt{2} adalah 2.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{2-1}$$

Kurangi 1 dari 2 untuk mendapatkan 1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{1}$$

Bilangan apa pun dapat dibagi menggunakan bilangan itu sendiri.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)$$

Terapkan properti distributif dengan mengalikan setiap suku \sqrt{2}+1 dengan setiap suku -\sqrt{2}-1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)$$

Nyatakan \frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right) sebagai pecahan tunggal.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Terapkan properti distributif dengan mengalikan setiap suku \sqrt{2}-1 dengan setiap suku 2\sqrt{2}-1.

$$\frac{\left(2\left(\sqrt{2}\right)^{2}-\sqrt{2}-2\sqrt{2}+1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Kuadrat \sqrt{2} adalah 2.

$$\frac{\left(2\times 2-\sqrt{2}-2\sqrt{2}+1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Kalikan 2 dan 2 untuk mendapatkan 4.

$$\frac{\left(4-\sqrt{2}-2\sqrt{2}+1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Gabungkan -\sqrt{2} dan -2\sqrt{2} untuk mendapatkan -3\sqrt{2}.

$$\frac{\left(4-3\sqrt{2}+1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Tambahkan 4 dan 1 untuk mendapatkan 5.

$$\frac{\left(5-3\sqrt{2}\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Terapkan properti distributif dengan mengalikan setiap suku 5-3\sqrt{2} dengan setiap suku \sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)-5\sqrt{2}+5\left(-\sqrt{2}\right)-5-3\left(-\sqrt{2}\right)\left(\sqrt{2}\right)^{2}+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Kuadrat \sqrt{2} adalah 2.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)-5\sqrt{2}+5\left(-\sqrt{2}\right)-5-3\left(-\sqrt{2}\right)\times 2+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Kalikan -3 dan 2 untuk mendapatkan -6.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)-5\sqrt{2}+5\left(-\sqrt{2}\right)-5-6\left(-\sqrt{2}\right)+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Kalikan -6 dan -1 untuk mendapatkan 6.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)-5\sqrt{2}+5\left(-\sqrt{2}\right)-5+6\sqrt{2}+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Gabungkan -5\sqrt{2} dan 6\sqrt{2} untuk mendapatkan \sqrt{2}.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)-5+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Kuadrat \sqrt{2} adalah 2.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)-5+3\times 2-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Kalikan 3 dan 2 untuk mendapatkan 6.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)-5+6-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Tambahkan -5 dan 6 untuk mendapatkan 1.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)+1-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Kalikan -3 dan -1 untuk mendapatkan 3.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)+1+3\sqrt{2}\sqrt{2}+3\sqrt{2}}{7}$$

Kalikan \sqrt{2} dan \sqrt{2} untuk mendapatkan 2.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)+1+3\times 2+3\sqrt{2}}{7}$$

Gabungkan \sqrt{2} dan 3\sqrt{2} untuk mendapatkan 4\sqrt{2}.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+4\sqrt{2}+5\left(-\sqrt{2}\right)+1+3\times 2}{7}$$

Kalikan 3 dan 2 untuk mendapatkan 6.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+4\sqrt{2}+5\left(-\sqrt{2}\right)+1+6}{7}$$

Tambahkan 1 dan 6 untuk mendapatkan 7.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+4\sqrt{2}+5\left(-\sqrt{2}\right)+7}{7}$$

Kalikan 5 dan -1 untuk mendapatkan -5.

$$\frac{-5\sqrt{2}\sqrt{2}+4\sqrt{2}+5\left(-1\right)\sqrt{2}+7}{7}$$

Kalikan \sqrt{2} dan \sqrt{2} untuk mendapatkan 2.

$$\frac{-5\times 2+4\sqrt{2}+5\left(-1\right)\sqrt{2}+7}{7}$$

Kalikan -5 dan 2 untuk mendapatkan -10.

$$\frac{-10+4\sqrt{2}+5\left(-1\right)\sqrt{2}+7}{7}$$

Kalikan 5 dan -1 untuk mendapatkan -5.

$$\frac{-10+4\sqrt{2}-5\sqrt{2}+7}{7}$$

Gabungkan 4\sqrt{2} dan -5\sqrt{2} untuk mendapatkan -\sqrt{2}.

$$\frac{-10-\sqrt{2}+7}{7}$$

Tambahkan -10 dan 7 untuk mendapatkan -3.

$$\frac{-3-\sqrt{2}}{7}$$