Solve frac{sqrt{6}-2sqrt{2}}{7-4sqrt{3}}*frac{sqrt{70}}{sqrt{2}+2}

$$\frac { \sqrt { 6 } - 2 \sqrt { 2 } } { 7 - 4 \sqrt { 3 } } \cdot \frac { \sqrt { 70 } } { \sqrt { 2 } + 2 }$$

Evaluate

$$\sqrt{210}+2\sqrt{70}-2\sqrt{105}-4\sqrt{35}\approx -12.933643387$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{6}-2\sqrt{2}}{7-4\sqrt{3}} by multiplying numerator and denominator by 7+4\sqrt{3}.

$$\frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)}{\left(7-4\sqrt{3}\right)\left(7+4\sqrt{3}\right)}\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Consider \left(7-4\sqrt{3}\right)\left(7+4\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}.

$$\frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)}{7^{2}-\left(-4\sqrt{3}\right)^{2}}\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)}{49-\left(-4\sqrt{3}\right)^{2}}\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Expand \left(-4\sqrt{3}\right)^{2}.

$$\frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)}{49-\left(-4\right)^{2}\left(\sqrt{3}\right)^{2}}\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

The square of \sqrt{3} is 3.

$$\frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)}{49-16\times 3}\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Multiply 16 and 3 to get 48.

$$\frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)}{49-48}\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Subtract 48 from 49 to get 1.

$$\frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)}{1}\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Anything divided by one gives itself.

$$\left(\sqrt{6}-2\sqrt{2}\right)\left(7+4\sqrt{3}\right)\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Apply the distributive property by multiplying each term of \sqrt{6}-2\sqrt{2} by each term of 7+4\sqrt{3}.

$$\left(7\sqrt{6}+4\sqrt{6}\sqrt{3}-14\sqrt{2}-8\sqrt{3}\sqrt{2}\right)\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\left(7\sqrt{6}+4\sqrt{3}\sqrt{2}\sqrt{3}-14\sqrt{2}-8\sqrt{3}\sqrt{2}\right)\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\left(7\sqrt{6}+4\times 3\sqrt{2}-14\sqrt{2}-8\sqrt{3}\sqrt{2}\right)\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Multiply 4 and 3 to get 12.

$$\left(7\sqrt{6}+12\sqrt{2}-14\sqrt{2}-8\sqrt{3}\sqrt{2}\right)\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Combine 12\sqrt{2} and -14\sqrt{2} to get -2\sqrt{2}.

$$\left(7\sqrt{6}-2\sqrt{2}-8\sqrt{3}\sqrt{2}\right)\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\left(7\sqrt{6}-2\sqrt{2}-8\sqrt{6}\right)\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Combine 7\sqrt{6} and -8\sqrt{6} to get -\sqrt{6}.

$$\left(-\sqrt{6}-2\sqrt{2}\right)\times \frac{\sqrt{70}}{\sqrt{2}+2}$$

Rationalize the denominator of \frac{\sqrt{70}}{\sqrt{2}+2} by multiplying numerator and denominator by \sqrt{2}-2.

$$\left(-\sqrt{6}-2\sqrt{2}\right)\times \frac{\sqrt{70}\left(\sqrt{2}-2\right)}{\left(\sqrt{2}+2\right)\left(\sqrt{2}-2\right)}$$

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

$$\left(-\sqrt{6}-2\sqrt{2}\right)\times \frac{\sqrt{70}\left(\sqrt{2}-2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}$$

Square \sqrt{2}. Square 2.

$$\left(-\sqrt{6}-2\sqrt{2}\right)\times \frac{\sqrt{70}\left(\sqrt{2}-2\right)}{2-4}$$

Subtract 4 from 2 to get -2.

$$\left(-\sqrt{6}-2\sqrt{2}\right)\times \frac{\sqrt{70}\left(\sqrt{2}-2\right)}{-2}$$

Express \left(-\sqrt{6}-2\sqrt{2}\right)\times \frac{\sqrt{70}\left(\sqrt{2}-2\right)}{-2} as a single fraction.

$$\frac{\left(-\sqrt{6}-2\sqrt{2}\right)\sqrt{70}\left(\sqrt{2}-2\right)}{-2}$$

Use the distributive property to multiply -\sqrt{6}-2\sqrt{2} by \sqrt{70}.

$$\frac{\left(-\sqrt{6}\sqrt{70}-2\sqrt{2}\sqrt{70}\right)\left(\sqrt{2}-2\right)}{-2}$$

To multiply \sqrt{6} and \sqrt{70}, multiply the numbers under the square root.

$$\frac{\left(-\sqrt{420}-2\sqrt{2}\sqrt{70}\right)\left(\sqrt{2}-2\right)}{-2}$$

Factor 70=2\times 35. Rewrite the square root of the product \sqrt{2\times 35} as the product of square roots \sqrt{2}\sqrt{35}.

$$\frac{\left(-\sqrt{420}-2\sqrt{2}\sqrt{2}\sqrt{35}\right)\left(\sqrt{2}-2\right)}{-2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{\left(-\sqrt{420}-2\times 2\sqrt{35}\right)\left(\sqrt{2}-2\right)}{-2}$$

Multiply -2 and 2 to get -4.

$$\frac{\left(-\sqrt{420}-4\sqrt{35}\right)\left(\sqrt{2}-2\right)}{-2}$$

Apply the distributive property by multiplying each term of -\sqrt{420}-4\sqrt{35} by each term of \sqrt{2}-2.

$$\frac{-\sqrt{420}\sqrt{2}+2\sqrt{420}-4\sqrt{35}\sqrt{2}+8\sqrt{35}}{-2}$$

Factor 420=2\times 210. Rewrite the square root of the product \sqrt{2\times 210} as the product of square roots \sqrt{2}\sqrt{210}.

$$\frac{-\sqrt{2}\sqrt{210}\sqrt{2}+2\sqrt{420}-4\sqrt{35}\sqrt{2}+8\sqrt{35}}{-2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{-2\sqrt{210}+2\sqrt{420}-4\sqrt{35}\sqrt{2}+8\sqrt{35}}{-2}$$

Factor 420=2^{2}\times 105. Rewrite the square root of the product \sqrt{2^{2}\times 105} as the product of square roots \sqrt{2^{2}}\sqrt{105}. Take the square root of 2^{2}.

$$\frac{-2\sqrt{210}+2\times 2\sqrt{105}-4\sqrt{35}\sqrt{2}+8\sqrt{35}}{-2}$$

Multiply 2 and 2 to get 4.

$$\frac{-2\sqrt{210}+4\sqrt{105}-4\sqrt{35}\sqrt{2}+8\sqrt{35}}{-2}$$

To multiply \sqrt{35} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{-2\sqrt{210}+4\sqrt{105}-4\sqrt{70}+8\sqrt{35}}{-2}$$

Divide each term of -2\sqrt{210}+4\sqrt{105}-4\sqrt{70}+8\sqrt{35} by -2 to get \sqrt{210}-2\sqrt{105}+2\sqrt{70}-4\sqrt{35}.

$$\sqrt{210}-2\sqrt{105}+2\sqrt{70}-4\sqrt{35}$$