Solve 10/(sqrt(3) + sqrt(3/3) + 1/(sqrt(3)) * 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

$$\frac{10}{\sqrt{3}+\sqrt{\frac{3}{3}}+\frac{1}{\sqrt{3}}\sqrt{3}}$$

Evaluate

$20-10\sqrt{3}\approx 2.679491924$

Short Solution Steps

Divide $3$ by $3$ to get $1$.

$$\frac{10}{\sqrt{3}+\sqrt{1}+\frac{1}{\sqrt{3}}\sqrt{3}}$$

Calculate the square root of $1$ and get $1$.

$$\frac{10}{\sqrt{3}+1+\frac{1}{\sqrt{3}}\sqrt{3}}$$

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

$$\frac{10}{\sqrt{3}+1+\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{3}}$$

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

$$\frac{10}{\sqrt{3}+1+\frac{\sqrt{3}}{3}\sqrt{3}}$$

Express $\frac{\sqrt{3}}{3}\sqrt{3}$ as a single fraction.

$$\frac{10}{\sqrt{3}+1+\frac{\sqrt{3}\sqrt{3}}{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $\sqrt{3}+1$ times $\frac{3}{3}$.

$$\frac{10}{\frac{3\left(\sqrt{3}+1\right)}{3}+\frac{\sqrt{3}\sqrt{3}}{3}}$$

Since $\frac{3\left(\sqrt{3}+1\right)}{3}$ and $\frac{\sqrt{3}\sqrt{3}}{3}$ have the same denominator, add them by adding their numerators.

$$\frac{10}{\frac{3\left(\sqrt{3}+1\right)+\sqrt{3}\sqrt{3}}{3}}$$

Do the multiplications in $3\left(\sqrt{3}+1\right)+\sqrt{3}\sqrt{3}$.

$$\frac{10}{\frac{3\sqrt{3}+3+3}{3}}$$

Do the calculations in $3\sqrt{3}+3+3$.

$$\frac{10}{\frac{3\sqrt{3}+6}{3}}$$

Divide each term of $3\sqrt{3}+6$ by $3$ to get $\sqrt{3}+2$.

$$\frac{10}{\sqrt{3}+2}$$

Rationalize the denominator of $\frac{10}{\sqrt{3}+2}$ by multiplying numerator and denominator by $\sqrt{3}-2$.

$$\frac{10\left(\sqrt{3}-2\right)}{\left(\sqrt{3}+2\right)\left(\sqrt{3}-2\right)}$$

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

$$\frac{10\left(\sqrt{3}-2\right)}{\left(\sqrt{3}\right)^{2}-2^{2}}$$

Square $\sqrt{3}$. Square $2$.

$$\frac{10\left(\sqrt{3}-2\right)}{3-4}$$

Subtract $4$ from $3$ to get $-1$.

$$\frac{10\left(\sqrt{3}-2\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-10\left(\sqrt{3}-2\right)$$

Use the distributive property to multiply $-10$ by $\sqrt{3}-2$.

$$-10\sqrt{3}+20$$