Solve frac{25-15sqrt{3}-15sqrt{3}+27}{25-13sqrt{3}-15sqrt{3}-27}

$$\frac { 25 - 15 \sqrt { 3 } - 15 \sqrt { 3 } + 27 } { 25 - 13 \sqrt { 3 } - 15 \sqrt { 3 } - 27 }$$

Evaluate

$$\frac{656-379\sqrt{3}}{587}\approx -0.000761935$$

Solution Steps

Combine -15\sqrt{3} and -15\sqrt{3} to get -30\sqrt{3}.

$$\frac{25-30\sqrt{3}+27}{25-13\sqrt{3}-15\sqrt{3}-27}$$

Add 25 and 27 to get 52.

$$\frac{52-30\sqrt{3}}{25-13\sqrt{3}-15\sqrt{3}-27}$$

Combine -13\sqrt{3} and -15\sqrt{3} to get -28\sqrt{3}.

$$\frac{52-30\sqrt{3}}{25-28\sqrt{3}-27}$$

Subtract 27 from 25 to get -2.

$$\frac{52-30\sqrt{3}}{-2-28\sqrt{3}}$$

Rationalize the denominator of \frac{52-30\sqrt{3}}{-2-28\sqrt{3}} by multiplying numerator and denominator by -2+28\sqrt{3}.

$$\frac{\left(52-30\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}{\left(-2-28\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}$$

Consider \left(-2-28\sqrt{3}\right)\left(-2+28\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}.

$$\frac{\left(52-30\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}{\left(-2\right)^{2}-\left(-28\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(52-30\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}{4-\left(-28\sqrt{3}\right)^{2}}$$

Expand \left(-28\sqrt{3}\right)^{2}.

$$\frac{\left(52-30\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}{4-\left(-28\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -28 to the power of 2 and get 784.

$$\frac{\left(52-30\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}{4-784\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(52-30\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}{4-784\times 3}$$

Multiply 784 and 3 to get 2352.

$$\frac{\left(52-30\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}{4-2352}$$

Subtract 2352 from 4 to get -2348.

$$\frac{\left(52-30\sqrt{3}\right)\left(-2+28\sqrt{3}\right)}{-2348}$$

Apply the distributive property by multiplying each term of 52-30\sqrt{3} by each term of -2+28\sqrt{3}.

$$\frac{-104+1456\sqrt{3}+60\sqrt{3}-840\left(\sqrt{3}\right)^{2}}{-2348}$$

Combine 1456\sqrt{3} and 60\sqrt{3} to get 1516\sqrt{3}.

$$\frac{-104+1516\sqrt{3}-840\left(\sqrt{3}\right)^{2}}{-2348}$$

The square of \sqrt{3} is 3.

$$\frac{-104+1516\sqrt{3}-840\times 3}{-2348}$$

Multiply -840 and 3 to get -2520.

$$\frac{-104+1516\sqrt{3}-2520}{-2348}$$

Subtract 2520 from -104 to get -2624.

$$\frac{-2624+1516\sqrt{3}}{-2348}$$