How to solve 3/(6 + 2sqrt(3))

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

Solve 3/(6 + 2sqrt(3)) | Equatix

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Question

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

Evaluate

$\frac{3-\sqrt{3}}{4}\approx 0.316987298$

Solution Steps

Rationalize the denominator of $\frac{3}{6+2\sqrt{3}}$ by multiplying numerator and denominator by $6-2\sqrt{3}$.

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

Consider $\left(6+2\sqrt{3}\right)\left(6-2\sqrt{3}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

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

Calculate $6$ to the power of $2$ and get $36$.

$$\frac{3\left(6-2\sqrt{3}\right)}{36-\left(2\sqrt{3}\right)^{2}}$$

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

$$\frac{3\left(6-2\sqrt{3}\right)}{36-2^{2}\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{3\left(6-2\sqrt{3}\right)}{36-4\left(\sqrt{3}\right)^{2}}$$

The square of $\sqrt{3}$ is $3$.

$$\frac{3\left(6-2\sqrt{3}\right)}{36-4\times 3}$$

Multiply $4$ and $3$ to get $12$.

$$\frac{3\left(6-2\sqrt{3}\right)}{36-12}$$

Subtract $12$ from $36$ to get $24$.

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

Divide $3\left(6-2\sqrt{3}\right)$ by $24$ to get $\frac{1}{8}\left(6-2\sqrt{3}\right)$.

$$\frac{1}{8}\left(6-2\sqrt{3}\right)$$

Use the distributive property to multiply $\frac{1}{8}$ by $6-2\sqrt{3}$.

$$\frac{1}{8}\times 6+\frac{1}{8}\left(-2\right)\sqrt{3}$$

Multiply $\frac{1}{8}$ and $6$ to get $\frac{6}{8}$.

$$\frac{6}{8}+\frac{1}{8}\left(-2\right)\sqrt{3}$$

Reduce the fraction $\frac{6}{8}$ to lowest terms by extracting and canceling out $2$.

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

Multiply $\frac{1}{8}$ and $-2$ to get $\frac{-2}{8}$.

$$\frac{3}{4}+\frac{-2}{8}\sqrt{3}$$

Reduce the fraction $\frac{-2}{8}$ to lowest terms by extracting and canceling out $2$.

$$\frac{3}{4}-\frac{1}{4}\sqrt{3}$$

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