Solve (sqrt(7) + sqrt(2))/(9 + 2sqrt(14)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{\sqrt{7}+\sqrt{2}}{9+2\sqrt{14}}$$

Evaluate

$\frac{\sqrt{7}-\sqrt{2}}{5}\approx 0.24630755$

Solution Steps

Rationalize the denominator of $\frac{\sqrt{7}+\sqrt{2}}{9+2\sqrt{14}}$ by multiplying numerator and denominator by $9-2\sqrt{14}$.

$$\frac{\left(\sqrt{7}+\sqrt{2}\right)\left(9-2\sqrt{14}\right)}{\left(9+2\sqrt{14}\right)\left(9-2\sqrt{14}\right)}$$

Consider $\left(9+2\sqrt{14}\right)\left(9-2\sqrt{14}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\left(\sqrt{7}+\sqrt{2}\right)\left(9-2\sqrt{14}\right)}{9^{2}-\left(2\sqrt{14}\right)^{2}}$$

Calculate $9$ to the power of $2$ and get $81$.

$$\frac{\left(\sqrt{7}+\sqrt{2}\right)\left(9-2\sqrt{14}\right)}{81-\left(2\sqrt{14}\right)^{2}}$$

Expand $\left(2\sqrt{14}\right)^{2}$.

$$\frac{\left(\sqrt{7}+\sqrt{2}\right)\left(9-2\sqrt{14}\right)}{81-2^{2}\left(\sqrt{14}\right)^{2}}$$

Calculate $2$ to the power of $2$ and get $4$.

$$\frac{\left(\sqrt{7}+\sqrt{2}\right)\left(9-2\sqrt{14}\right)}{81-4\left(\sqrt{14}\right)^{2}}$$

The square of $\sqrt{14}$ is $14$.

$$\frac{\left(\sqrt{7}+\sqrt{2}\right)\left(9-2\sqrt{14}\right)}{81-4\times 14}$$

Multiply $4$ and $14$ to get $56$.

$$\frac{\left(\sqrt{7}+\sqrt{2}\right)\left(9-2\sqrt{14}\right)}{81-56}$$

Subtract $56$ from $81$ to get $25$.

$$\frac{\left(\sqrt{7}+\sqrt{2}\right)\left(9-2\sqrt{14}\right)}{25}$$

Apply the distributive property by multiplying each term of $\sqrt{7}+\sqrt{2}$ by each term of $9-2\sqrt{14}$.

$$\frac{9\sqrt{7}-2\sqrt{7}\sqrt{14}+9\sqrt{2}-2\sqrt{2}\sqrt{14}}{25}$$

Factor $14=7\times 2$. Rewrite the square root of the product $\sqrt{7\times 2}$ as the product of square roots $\sqrt{7}\sqrt{2}$.

$$\frac{9\sqrt{7}-2\sqrt{7}\sqrt{7}\sqrt{2}+9\sqrt{2}-2\sqrt{2}\sqrt{14}}{25}$$

Multiply $\sqrt{7}$ and $\sqrt{7}$ to get $7$.

$$\frac{9\sqrt{7}-2\times 7\sqrt{2}+9\sqrt{2}-2\sqrt{2}\sqrt{14}}{25}$$

Multiply $-2$ and $7$ to get $-14$.

$$\frac{9\sqrt{7}-14\sqrt{2}+9\sqrt{2}-2\sqrt{2}\sqrt{14}}{25}$$

Combine $-14\sqrt{2}$ and $9\sqrt{2}$ to get $-5\sqrt{2}$.

$$\frac{9\sqrt{7}-5\sqrt{2}-2\sqrt{2}\sqrt{14}}{25}$$

Factor $14=2\times 7$. Rewrite the square root of the product $\sqrt{2\times 7}$ as the product of square roots $\sqrt{2}\sqrt{7}$.

$$\frac{9\sqrt{7}-5\sqrt{2}-2\sqrt{2}\sqrt{2}\sqrt{7}}{25}$$

Multiply $\sqrt{2}$ and $\sqrt{2}$ to get $2$.

$$\frac{9\sqrt{7}-5\sqrt{2}-2\times 2\sqrt{7}}{25}$$

Multiply $-2$ and $2$ to get $-4$.

$$\frac{9\sqrt{7}-5\sqrt{2}-4\sqrt{7}}{25}$$

Combine $9\sqrt{7}$ and $-4\sqrt{7}$ to get $5\sqrt{7}$.

$$\frac{5\sqrt{7}-5\sqrt{2}}{25}$$