Solve 326/7-4sqrt{3}*frac{sqrt{6}-2sqrt{2}}{sqrt{3}+2}

$$\frac { 326 } { 7 - 4 \sqrt { 3 } } \cdot \frac { \sqrt { 6 } - 2 \sqrt { 2 } } { \sqrt { 3 } + 2 }$$

Evaluate

$$-326\sqrt{2}\approx -461.033621334$$

Solution Steps

Rationalize the denominator of \frac{326}{7-4\sqrt{3}} by multiplying numerator and denominator by 7+4\sqrt{3}.

$$\frac{326\left(7+4\sqrt{3}\right)}{\left(7-4\sqrt{3}\right)\left(7+4\sqrt{3}\right)}\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

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}.

$$\frac{326\left(7+4\sqrt{3}\right)}{7^{2}-\left(-4\sqrt{3}\right)^{2}}\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{326\left(7+4\sqrt{3}\right)}{49-\left(-4\sqrt{3}\right)^{2}}\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

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

$$\frac{326\left(7+4\sqrt{3}\right)}{49-\left(-4\right)^{2}\left(\sqrt{3}\right)^{2}}\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

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

$$\frac{326\left(7+4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

The square of \sqrt{3} is 3.

$$\frac{326\left(7+4\sqrt{3}\right)}{49-16\times 3}\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

Multiply 16 and 3 to get 48.

$$\frac{326\left(7+4\sqrt{3}\right)}{49-48}\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

Subtract 48 from 49 to get 1.

$$\frac{326\left(7+4\sqrt{3}\right)}{1}\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

Anything divided by one gives itself.

$$326\left(7+4\sqrt{3}\right)\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

Use the distributive property to multiply 326 by 7+4\sqrt{3}.

$$\left(2282+1304\sqrt{3}\right)\times \frac{\sqrt{6}-2\sqrt{2}}{\sqrt{3}+2}$$

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

$$\left(2282+1304\sqrt{3}\right)\times \frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{3}+2\right)\left(\sqrt{3}-2\right)}$$

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}.

$$\left(2282+1304\sqrt{3}\right)\times \frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{3}\right)^{2}-2^{2}}$$

Square \sqrt{3}. Square 2.

$$\left(2282+1304\sqrt{3}\right)\times \frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)}{3-4}$$

Subtract 4 from 3 to get -1.

$$\left(2282+1304\sqrt{3}\right)\times \frac{\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$\left(2282+1304\sqrt{3}\right)\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

Use the distributive property to multiply 2282+1304\sqrt{3} by -\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right).

$$2282\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

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

$$2282\left(-\left(\sqrt{6}\sqrt{3}-2\sqrt{6}-2\sqrt{2}\sqrt{3}+4\sqrt{2}\right)\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

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}.

$$2282\left(-\left(\sqrt{3}\sqrt{2}\sqrt{3}-2\sqrt{6}-2\sqrt{2}\sqrt{3}+4\sqrt{2}\right)\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

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

$$2282\left(-\left(3\sqrt{2}-2\sqrt{6}-2\sqrt{2}\sqrt{3}+4\sqrt{2}\right)\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

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

$$2282\left(-\left(3\sqrt{2}-2\sqrt{6}-2\sqrt{6}+4\sqrt{2}\right)\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

Combine -2\sqrt{6} and -2\sqrt{6} to get -4\sqrt{6}.

$$2282\left(-\left(3\sqrt{2}-4\sqrt{6}+4\sqrt{2}\right)\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

Combine 3\sqrt{2} and 4\sqrt{2} to get 7\sqrt{2}.

$$2282\left(-\left(7\sqrt{2}-4\sqrt{6}\right)\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

To find the opposite of 7\sqrt{2}-4\sqrt{6}, find the opposite of each term.

$$2282\left(-7\sqrt{2}-\left(-4\sqrt{6}\right)\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

The opposite of -4\sqrt{6} is 4\sqrt{6}.

$$2282\left(-7\sqrt{2}+4\sqrt{6}\right)+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

Use the distributive property to multiply 2282 by -7\sqrt{2}+4\sqrt{6}.

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-\left(\sqrt{6}-2\sqrt{2}\right)\left(\sqrt{3}-2\right)\right)$$

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

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-\left(\sqrt{6}\sqrt{3}-2\sqrt{6}-2\sqrt{2}\sqrt{3}+4\sqrt{2}\right)\right)$$

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}.

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-\left(\sqrt{3}\sqrt{2}\sqrt{3}-2\sqrt{6}-2\sqrt{2}\sqrt{3}+4\sqrt{2}\right)\right)$$

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

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-\left(3\sqrt{2}-2\sqrt{6}-2\sqrt{2}\sqrt{3}+4\sqrt{2}\right)\right)$$

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

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-\left(3\sqrt{2}-2\sqrt{6}-2\sqrt{6}+4\sqrt{2}\right)\right)$$

Combine -2\sqrt{6} and -2\sqrt{6} to get -4\sqrt{6}.

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-\left(3\sqrt{2}-4\sqrt{6}+4\sqrt{2}\right)\right)$$

Combine 3\sqrt{2} and 4\sqrt{2} to get 7\sqrt{2}.

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-\left(7\sqrt{2}-4\sqrt{6}\right)\right)$$

To find the opposite of 7\sqrt{2}-4\sqrt{6}, find the opposite of each term.

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-7\sqrt{2}-\left(-4\sqrt{6}\right)\right)$$

The opposite of -4\sqrt{6} is 4\sqrt{6}.

$$-15974\sqrt{2}+9128\sqrt{6}+1304\sqrt{3}\left(-7\sqrt{2}+4\sqrt{6}\right)$$

Use the distributive property to multiply 1304\sqrt{3} by -7\sqrt{2}+4\sqrt{6}.

$$-15974\sqrt{2}+9128\sqrt{6}-9128\sqrt{3}\sqrt{2}+5216\sqrt{3}\sqrt{6}$$

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

$$-15974\sqrt{2}+9128\sqrt{6}-9128\sqrt{6}+5216\sqrt{3}\sqrt{6}$$

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}.

$$-15974\sqrt{2}+9128\sqrt{6}-9128\sqrt{6}+5216\sqrt{3}\sqrt{3}\sqrt{2}$$

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

$$-15974\sqrt{2}+9128\sqrt{6}-9128\sqrt{6}+5216\times 3\sqrt{2}$$

Multiply 5216 and 3 to get 15648.

$$-15974\sqrt{2}+9128\sqrt{6}-9128\sqrt{6}+15648\sqrt{2}$$

Combine 9128\sqrt{6} and -9128\sqrt{6} to get 0.

$$-15974\sqrt{2}+15648\sqrt{2}$$

Combine -15974\sqrt{2} and 15648\sqrt{2} to get -326\sqrt{2}.

$$-326\sqrt{2}$$