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\frac{4\sqrt{2}+\sqrt{48}}{\sqrt{8}+\sqrt{12}}
Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.
\frac{4\sqrt{2}+4\sqrt{3}}{\sqrt{8}+\sqrt{12}}
Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.
\frac{4\sqrt{2}+4\sqrt{3}}{2\sqrt{2}+\sqrt{12}}
Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.
\frac{4\sqrt{2}+4\sqrt{3}}{2\sqrt{2}+2\sqrt{3}}
Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{\left(2\sqrt{2}+2\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}
Rationalize the denominator of \frac{4\sqrt{2}+4\sqrt{3}}{2\sqrt{2}+2\sqrt{3}} by multiplying numerator and denominator by 2\sqrt{2}-2\sqrt{3}.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{\left(2\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}}
Consider \left(2\sqrt{2}+2\sqrt{3}\right)\left(2\sqrt{2}-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(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{2^{2}\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}}
Expand \left(2\sqrt{2}\right)^{2}.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{4\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}}
Calculate 2 to the power of 2 and get 4.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{4\times 2-\left(2\sqrt{3}\right)^{2}}
The square of \sqrt{2} is 2.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{8-\left(2\sqrt{3}\right)^{2}}
Multiply 4 and 2 to get 8.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{8-2^{2}\left(\sqrt{3}\right)^{2}}
Expand \left(2\sqrt{3}\right)^{2}.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{8-4\left(\sqrt{3}\right)^{2}}
Calculate 2 to the power of 2 and get 4.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{8-4\times 3}
The square of \sqrt{3} is 3.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{8-12}
Multiply 4 and 3 to get 12.
\frac{\left(4\sqrt{2}+4\sqrt{3}\right)\left(2\sqrt{2}-2\sqrt{3}\right)}{-4}
Subtract 12 from 8 to get -4.
\frac{8\left(\sqrt{2}\right)^{2}-8\sqrt{3}\sqrt{2}+8\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{-4}
Apply the distributive property by multiplying each term of 4\sqrt{2}+4\sqrt{3} by each term of 2\sqrt{2}-2\sqrt{3}.
\frac{8\times 2-8\sqrt{3}\sqrt{2}+8\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{-4}
The square of \sqrt{2} is 2.
\frac{16-8\sqrt{3}\sqrt{2}+8\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{-4}
Multiply 8 and 2 to get 16.
\frac{16-8\sqrt{6}+8\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{-4}
To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.
\frac{16-8\sqrt{6}+8\sqrt{6}-8\left(\sqrt{3}\right)^{2}}{-4}
To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.
\frac{16-8\left(\sqrt{3}\right)^{2}}{-4}
Combine -8\sqrt{6} and 8\sqrt{6} to get 0.
\frac{16-8\times 3}{-4}
The square of \sqrt{3} is 3.
\frac{16-24}{-4}
Multiply -8 and 3 to get -24.
\frac{-8}{-4}
Subtract 24 from 16 to get -8.
2
Divide -8 by -4 to get 2.