To add or subtract radicals, they must be the same root and have the same number under the radical before combining them. Look for perfect squares that divide into the number under the radicals because those can be simplified.

\(\displaystyle -\sqrt{24}+6\sqrt{40}+2\sqrt{80}\)

\(\displaystyle -\sqrt{4\cdot 6}+6\sqrt{4\cdot 10}+2\sqrt{16\cdot 5}\)

Take the square roots of each of the perfect squares in these radicals and bring it out of the radical. It will multiply to any coefficient in front of that radical

\(\displaystyle -2\sqrt{6}+6\cdot 2\sqrt{10}+2\cdot 4\sqrt{5}\)

\(\displaystyle -2\sqrt{6}+12\sqrt{10}+8\sqrt{5}\)

To add or subtract radicals, they must be the same root and have the same number under the radical before combining them. Look for perfect squares that divide into the number under the radicals because those can be simplified.

\(\displaystyle 4\sqrt{169}+7\sqrt{52}+5\sqrt{325}\)

\(\displaystyle 4\cdot 13+7\sqrt{4\cdot 13}+5\sqrt{25\cdot 13}\)

Take the square roots of each of the perfect squares in these radicals and bring it out of the radical. It will multiply to any coefficient in front of that radical

\(\displaystyle 52+7\cdot 2\sqrt{13}+5\cdot 5\sqrt{13}\)

\(\displaystyle 52+14\sqrt{13}+25\sqrt{13}\)

\(\displaystyle 52+39\sqrt{13}\)

To add or subtract radicals, they must be the same root and have the same number under the radical before combining them. Look for perfect squares that divide into the number under the radicals because those can be simplified.

\(\displaystyle 9\sqrt{14}-3\sqrt{126}+5\sqrt{896}\)

\(\displaystyle 9\sqrt{14}-3\sqrt{9\cdot 14}+5\sqrt{64\cdot 14}\)

Take the square roots of each of the perfect squares in these radicals and bring it out of the radical. It will multiply to any coefficient in front of that radical

\(\displaystyle 9\sqrt{14}-3\cdot 3\sqrt{14}+5\cdot 8\sqrt{14}\)

\(\displaystyle 9\sqrt{14}-9\sqrt{14}+40\sqrt{14}\)

\(\displaystyle 40\sqrt{14}\)

To add or subtract radicals, they must be the same root and have the same number under the radical before combining them. Look for perfect squares that divide into the number under the radicals because those can be simplified.

\(\displaystyle 10\sqrt{44}+5\sqrt{99}-7\sqrt{275}\)

\(\displaystyle 10\sqrt{4\cdot 11}+5\sqrt{9\cdot 11}-7\sqrt{25\cdot 11}\)

Take the square roots of each of the perfect squares in these radicals and bring it out of the radical. It will multiply to any coefficient in front of that radical

\(\displaystyle 10\cdot 2\sqrt{11}+5\cdot 3\sqrt{11}-7\cdot 5\sqrt{11}\)

\(\displaystyle 20\sqrt{11}+15\sqrt{11}-35\sqrt{11}\)

\(\displaystyle 0\)

Rewrite \(\displaystyle \sqrt{12}\) and \(\displaystyle \sqrt{24}\) by their factors.  The first two terms are already in their simplest forms.

\(\displaystyle \sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt3\)

\(\displaystyle \sqrt{24} = \sqrt{4}\cdot \sqrt{6} = 2\sqrt6\)

Rewrite the expression.

\(\displaystyle \sqrt2+\sqrt3+\sqrt{12}+\sqrt{24}= \sqrt2+\sqrt3+ 2\sqrt3+2\sqrt6\)

Combine like-terms.

The answer is:  \(\displaystyle \sqrt2+3\sqrt3+2\sqrt6\)

Simplify the square roots by writing them as a common factor of perfect squares.

\(\displaystyle \sqrt{20} +\sqrt{45} = \sqrt4 \cdot \sqrt5+ \sqrt9 \cdot \sqrt5\)

Simplify the perfect squares.

\(\displaystyle \sqrt4 \cdot \sqrt5+ \sqrt9 \cdot \sqrt5 = 2\sqrt5+3\sqrt5\)

Combine like-terms.

The answer is:  \(\displaystyle 5\sqrt5\)

To add or subtract radicals, they must be the same root and have the same number under the radical before combining them. Look for perfect squares that divide into the number under the radicals because those can be simplified.

\(\displaystyle 3\sqrt{175}-4\sqrt{252}+2\sqrt{112}\)

\(\displaystyle 3\sqrt{25\cdot 7}-4\sqrt{36\cdot 7}+2\sqrt{16\cdot 7}\)

Take the square roots of each of the perfect squares in these radicals and bring it out of the radical. It will multiply to any coefficient in front of that radical

\(\displaystyle 3\cdot 5\sqrt{7}-4\cdot 6\sqrt{7}+2\cdot 4\sqrt{7}\)

\(\displaystyle 15\sqrt{7}-24\sqrt{7}+8\sqrt{7}\)

\(\displaystyle -\sqrt{7}\)

To add or subtract radicals, they must be the same root and have the same number under the radical before combining them. Look for perfect squares that divide into the number under the radicals because those can be simplified.

\(\displaystyle 3\sqrt{48}-\sqrt{72}+5\sqrt{75}\)

\(\displaystyle 3\sqrt{16\cdot 3}-\sqrt{36\cdot 2}+5\sqrt{25\cdot 3}\)

Take the square roots of each of the perfect squares in these radicals and bring it out of the radical. It will multiply to any coefficient in front of that radical

\(\displaystyle 3\cdot 4\sqrt{3}-6\sqrt{2}+5\cdot 5\sqrt{3}\)

\(\displaystyle 12\sqrt{3}-6\sqrt{2}+25\sqrt{3}\)

Remember, only radicals with the same number can be combined

\(\displaystyle 37\sqrt{3}-6\sqrt{2}\)

This is the final answer.

To add or subtract radicals, they must be the same root and have the same number under the radical before combining them. Look for perfect squares that divide into the number under the radicals because those can be simplified.

\(\displaystyle 2\sqrt{125}-\sqrt{320}+4\sqrt{180}\)

\(\displaystyle 2\sqrt{25\cdot 5}-\sqrt{64\cdot 5}+4\sqrt{36\cdot 5}\)

Take the square roots of each of the perfect squares in these radicals and bring it out of the radical. It will multiply to any coefficient in front of that radical

\(\displaystyle 2\cdot 5\sqrt{5}-8\sqrt{5}+4\cdot 6\sqrt{5}\)

\(\displaystyle 10\sqrt{5}-8\sqrt{5}+24\sqrt{5}\)

\(\displaystyle 26\sqrt{5}\)

Use common factors to simplify both radicals.

\(\displaystyle \sqrt{50}+\sqrt{125} = \sqrt{25\times 2}+\sqrt{25\times 5}\)

\(\displaystyle = \sqrt{25}\cdot \sqrt{2}+\sqrt{25}\cdot \sqrt{5}\)

Simplify the square roots.

The answer is:  \(\displaystyle 5\sqrt{2}+5\sqrt5\)