Solve (sqrt(2) + 1/(sqrt(3)))(sqrt(2) - 1/(sqrt(3))) | 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

$$(\sqrt{2}+\frac{1}{\sqrt{3}})(\sqrt{2}-\frac{1}{\sqrt{3}})$$

Evaluate

$\frac{5}{3}\approx 1.666666667$

Short Solution Steps

Rationalize the denominator of $\frac{1}{\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{3}$.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply $\sqrt{2}$ times $\frac{3}{3}$.

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

Since $\frac{3\sqrt{2}}{3}$ and $\frac{\sqrt{3}}{3}$ have the same denominator, add them by adding their numerators.

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

Rationalize the denominator of $\frac{1}{\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{3}$.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply $\sqrt{2}$ times $\frac{3}{3}$.

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

Since $\frac{3\sqrt{2}}{3}$ and $\frac{\sqrt{3}}{3}$ have the same denominator, subtract them by subtracting their numerators.

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

Multiply $\frac{3\sqrt{2}+\sqrt{3}}{3}$ times $\frac{3\sqrt{2}-\sqrt{3}}{3}$ by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

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

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

The square of $\sqrt{2}$ is $2$.

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

Multiply $9$ and $2$ to get $18$.

$$\frac{18-\left(\sqrt{3}\right)^{2}}{3\times 3}$$

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

$$\frac{18-3}{3\times 3}$$

Subtract $3$ from $18$ to get $15$.

$$\frac{15}{3\times 3}$$

Multiply $3$ and $3$ to get $9$.

$$\frac{15}{9}$$

Reduce the fraction $\frac{15}{9}$ to lowest terms by extracting and canceling out $3$.

$$\frac{5}{3}$$

Factor

$\frac{5}{3} = 1\frac{2}{3} = 1.6666666666666667$