In order to add or subtract, we need like radicals. This means radicals that have the same index AND the same radicand (expression under radical sign). However, the radicals need to be in simplified form before checking radicands. Once all radicals are in simpliest form, then just add or subtract the outside numbers.
Example Add or subtract if possible: \(\text{a) } 3\sqrt{5}+7\sqrt{5}\) \(\text{b) } 2\sqrt[3]{11}-\sqrt[3]{11}+3\sqrt[3]{88}\) \(\text{c) } -2\sqrt[4]{32x}-7\sqrt[4]{243x}\)
Pay attention to the index used in the problem. Our brain's think about square roots very well because we have been exposed to them intently. This next example has cube roots, or index 3. That means the focus is on numbers like \({1, 8=(2)^3, 27=(3)^3, 64=(4)^3, 125=(5)^3,\cdot\cdot\cdot}\).
Example Add \(\sqrt[3]{p^4q^7}-\sqrt[3]{64pq}\).
This last example incorporates fractions and radicals. This one has some tricky points, so go over this carefully
Example Subtract \(3\sqrt{\frac{8}{9}}-2\frac{\sqrt{27}}{\sqrt{108}}\).