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Add and subtract like radicals

Like radicals have the same index and the same radicand after simplification, so their coefficients can be combined by addition or subtraction while the radical part remains unchanged; for example, 35+25=553\sqrt5+2\sqrt5=5\sqrt5 and 12+3=33\sqrt{12}+\sqrt3=3\sqrt3. Unlike radicals such as 2\sqrt2 and 3\sqrt3 cannot be combined, and their radicands are not added; the scope is exact real-number expressions, chiefly square roots, rather than advanced symbolic nnth-root generalizations.

Detailed Explanation: Add and subtract like radicals

Like radicals have the same radical part after simplifying. To add or subtract them, combine only their coefficients and keep the radical part unchanged.

For example, simplify:

212+327−32\sqrt{12}+3\sqrt{27}-\sqrt{3}

Step 1: Simplify each radical.

Factor out perfect squares:

12=4⋅3=23\sqrt{12}=\sqrt{4\cdot3}=2\sqrt3 27=9⋅3=33\sqrt{27}=\sqrt{9\cdot3}=3\sqrt3

Substitute these into the expression:

2(23)+3(33)−32(2\sqrt3)+3(3\sqrt3)-\sqrt3

Step 2: Multiply the coefficients.

43+93−34\sqrt3+9\sqrt3-\sqrt3

Remember that 3\sqrt3 has an invisible coefficient of 11:

43+93−134\sqrt3+9\sqrt3-1\sqrt3

Step 3: Combine the coefficients.

4+9−1=124+9-1=12

Therefore,

123\boxed{12\sqrt3}

Do not add the radicands. For example, 2+3\sqrt2+\sqrt3 cannot be combined because the radicals are unlike.

Learn by doing: Add and subtract like radicals

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    Radicals - Adding and Subtracting (Values and Variables)

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