Solve 19/(7 + sqrt(5) * 3) + (7 - sqrt(53))/(7 - sqrt(53)) | 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

$$\frac{19}{7+\sqrt{5}3}+\frac{7-\sqrt{53}}{7-\sqrt{53}}$$

Evaluate

$\frac{137-57\sqrt{5}}{4}\approx 2.386031321$

Short Solution Steps

Divide $7-\sqrt{53}$ by $7-\sqrt{53}$ to get $1$.

$$\frac{19}{7+3\sqrt{5}}+1$$

Rationalize the denominator of $\frac{19}{7+3\sqrt{5}}$ by multiplying numerator and denominator by $7-3\sqrt{5}$.

$$\frac{19\left(7-3\sqrt{5}\right)}{\left(7+3\sqrt{5}\right)\left(7-3\sqrt{5}\right)}+1$$

Consider $\left(7+3\sqrt{5}\right)\left(7-3\sqrt{5}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{19\left(7-3\sqrt{5}\right)}{7^{2}-\left(3\sqrt{5}\right)^{2}}+1$$

Calculate $7$ to the power of $2$ and get $49$.

$$\frac{19\left(7-3\sqrt{5}\right)}{49-\left(3\sqrt{5}\right)^{2}}+1$$

Expand $\left(3\sqrt{5}\right)^{2}$.

$$\frac{19\left(7-3\sqrt{5}\right)}{49-3^{2}\left(\sqrt{5}\right)^{2}}+1$$

Calculate $3$ to the power of $2$ and get $9$.

$$\frac{19\left(7-3\sqrt{5}\right)}{49-9\left(\sqrt{5}\right)^{2}}+1$$

The square of $\sqrt{5}$ is $5$.

$$\frac{19\left(7-3\sqrt{5}\right)}{49-9\times 5}+1$$

Multiply $9$ and $5$ to get $45$.

$$\frac{19\left(7-3\sqrt{5}\right)}{49-45}+1$$

Subtract $45$ from $49$ to get $4$.

$$\frac{19\left(7-3\sqrt{5}\right)}{4}+1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $1$ times $\frac{4}{4}$.

$$\frac{19\left(7-3\sqrt{5}\right)}{4}+\frac{4}{4}$$

Since $\frac{19\left(7-3\sqrt{5}\right)}{4}$ and $\frac{4}{4}$ have the same denominator, add them by adding their numerators.

$$\frac{19\left(7-3\sqrt{5}\right)+4}{4}$$

Do the multiplications in $19\left(7-3\sqrt{5}\right)+4$.

$$\frac{133-57\sqrt{5}+4}{4}$$

Do the calculations in $133-57\sqrt{5}+4$.

$$\frac{137-57\sqrt{5}}{4}$$