Addition and Subtraction of Radicals
Two or more radicals can be added or subtracted only when they are similar radicals, that is, when they have the same index and the same radicand. $$ a \sqrt[n]{r} + b \sqrt[n]{r} = (a+b) \sqrt[n]{r} $$ $$ a \sqrt[n]{r} - b \sqrt[n]{r} = (a-b) \sqrt[n]{r} $$
The algebraic sum of two like radicals is a radical whose coefficient is the algebraic sum of the coefficients \(a\) and \(b\).
$$ a \sqrt[n]{r} + b \sqrt[n]{r} = (a+b) \sqrt[n]{r} $$
Example. Consider the sum of two like radicals. $$ 2 \sqrt{2} + 3 \sqrt{2} $$ Factor out the common radical and add the coefficients. $$ (2+3) \sqrt{2} = 5 \sqrt{2} $$
In general, even when the index of the root is the same, if the radicands are different the sum of the radicals is not equal to the radical of the sum of the radicands.
$$ \sqrt[n]{p} + \sqrt[n]{q} \ne \sqrt[n]{p+q} $$
The same observation applies to subtraction.
$$ \sqrt[n]{p} - \sqrt[n]{q} \ne \sqrt[n]{p-q} $$
Proof. Consider two radicals with the same index but different radicands. Check whether the sum of the radicals is equal to the radical of the sum of the radicands. $$ \sqrt{9}+\sqrt{16} = \sqrt{9+16} $$ Carry out the calculations. $$ \sqrt{3^2}+\sqrt{4^2} = \sqrt{25} $$ $$ 3+4 = \sqrt{5^2} $$ $$ 7 = 5 $$ The two expressions have different values. Therefore, in general, the sum of radicals is not equal to the radical of the sum of the radicands. This does not rule out the possibility that in exceptional cases the two expressions may coincide, but in general the rule does not hold.
A Practical Example
Consider two like radicals.
$$ 2 \sqrt{3} + 3 \sqrt{3} $$
Since the radicals are like radicals, their coefficients can be added.
$$ (2+3) \sqrt{3} $$
$$ 5 \sqrt{3} $$
Verification. Verify that the two expressions are equal. $$ 2 \sqrt{3} + 3 \sqrt{3} = 5 \sqrt{3} $$ The square root of 3 is approximately 1.73. $$ \sqrt{3}=1.73 $$ Substitute this approximate value for the square root of 3. $$ 2 \cdot 1.73 + 3 \cdot 1.73 = 5 \cdot 1.73 $$ $$ 3.46 + 5.19 = 8.65 $$ $$ 8.65 = 8.65 $$ Both expressions produce the same result. Therefore, the equality holds.
And so on.
