Addition and subtraction of radicals
The addition and subtraction of radicals is similar to the addition and subtraction of
algebraic expressions with like terms.
Notice how the distributive property is used to combine 6x and 2x.
6x + 2x = (6 + 2)x = 8x
The distributive property can also be used to add and subtract expressions containing radicals.
$$
3\sqrt{2} + 7 \sqrt{2} = (3 + 7)\sqrt{2} = 10 \sqrt{2} $$
$$
We \ call \ 3\sqrt{2}\ and \ 7 \sqrt{2} \ like\ radicals $$
Like radicals
Like radicals are radical expressions that have the same radicand and the same index.
Only like radicals can be added or subtracted using the distributive property. The following radical expressions cannot be added or subtracted.
$$ a. \
3\sqrt{2} + 2 \sqrt{3} $$
$$ b. \
8\sqrt[3]{3} + 5 \sqrt{3} $$
a. has the same index, but the radicand is not the same.
b. has the same radicand, but the index is not the same.
More examples of addition and subtraction of radicals
$$
6\sqrt{5} + 3 \sqrt{5} = (6 + 3)\sqrt{5} = 9 \sqrt{5} $$
$$
18\sqrt[3]{7}  5 \sqrt[3]{7} \sqrt[3]{7} = (18  5  1)\sqrt[3]{7} $$
$$
18\sqrt[3]{7}  5 \sqrt[3]{7}  \sqrt[3]{7} = 12\sqrt[3]{7} $$
Sometimes, you may need to simplify each radical until you get the same radicand before you add and / or subtract radicals. The next example demonstrates how.
$$
\sqrt{50} + \sqrt{8} = \sqrt{25 \times 2} + \sqrt{4 \times 2} $$
$$
\sqrt{50} + \sqrt{8} = \sqrt{25} \times \sqrt{ 2} + \sqrt{4} \times \sqrt{2} $$
$$
\sqrt{50} + \sqrt{8} = 5\sqrt{ 2} + 2\sqrt{2} $$
$$
\sqrt{50} + \sqrt{8} = (5 + 2) \sqrt{ 2} $$
$$
\sqrt{50} + \sqrt{8} = 7 \sqrt{ 2} $$

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