A radical expression that contains positive terms and negative terms have ways to be further simplified.

When a radical in an expression like such is presented: √4a + √16a – √9a, a ≥ 0

This is a radical expression because there is no equal sign and it includes radicals.

Since this expression has + and – symbols, we are able to combine like terms.

Combining like terms is done when the index of the root (4∛8, 3 is the index) is the same. The coefficient (4∛8, 4 is the coefficient) is the number that changes when the terms are combined and the radicand (4∛8, 8 is the radicand) stays the same.

Lastly, we need an understanding of simplifying radicals to help us simplify terms to create the same radicand and combine them.

√4a can be simplified to 2√a because √4 = 2

√16a can be simplified to 4√a because √16 = 4

– √9a can be simplified to – 3√a because √9 = 3, 3 x – = -3

√4a + √16a – √9a

2√a +  4√a – 3√a

Since the index and the radicands are the same, we are able to combine them to simplify the expression

3√a, a ≥ 0

 

Another example would be: 5e√24e³ – 7√54e⁵ + e²√6e + 6e, e ≥ 0

Each term is first simplified by finding the perfect square inside the radical and removing it. When it is removed from the radical, it is also rooted by the index (2√12
2√4⋅3
2√4⋅√3
2(2)√3
4√3)

5e√24e³ – 7√54e⁵ + e²√6e + 6e, e ≥ 0

5e√4⋅6e²⁻¹ – 7√9⋅6e²⁺²⁺¹ + e²√6e + 6e

5(2)e¹⁺¹√6e – 7(3)√6e²⁺²⁺¹ + e²√6e + 6e

10e²√6e – 21e²√6e + e²√6e + 6e

Then, since the index and the radicand is the same, the terms can be combined and simplified

-10e²√6e + 6e, e ≥ 0