How to solve (2)/(3-sqrt(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 (2)/(3-sqrt(3)) | Equatix

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Question

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

Evaluate

$\frac{\sqrt{3}}{3}+1\approx 1.577350269$

Solution Steps

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

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

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

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

Square $3$. Square $\sqrt{3}$.

$$\frac{2\left(3+\sqrt{3}\right)}{9-3}$$

Subtract $3$ from $9$ to get $6$.

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

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

$$\frac{1}{3}\left(3+\sqrt{3}\right)$$

Use the distributive property to multiply $\frac{1}{3}$ by $3+\sqrt{3}$.

$$\frac{1}{3}\times 3+\frac{1}{3}\sqrt{3}$$

Cancel out $3$ and $3$.

$$1+\frac{1}{3}\sqrt{3}$$

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