Adding Like Radicals

Simplify: $4\sqrt{3} + 9\sqrt{3}$

Check if radicals are like: Both have $\sqrt{3}$ - same index (2) and same radicand (3) = Like radicals

Add the coefficients: $4 + 9 = 13$ = 13

Keep the radical part: $13 \cdot \sqrt{3}$ = $13\sqrt{3}$

Answer: $13\sqrt{3}$

Subtracting Like Radicals

Simplify: $8\sqrt{7} - 3\sqrt{7}$

Check if radicals are like: Both have $\sqrt{7}$ - they are like radicals = Like radicals

Subtract the coefficients: $8 - 3 = 5$ = 5

Keep the radical part: $5 \cdot \sqrt{7}$ = $5\sqrt{7}$

Answer: $5\sqrt{7}$

Simplify First, Then Add

Simplify: $\sqrt{12} + \sqrt{27}$

Simplify $\sqrt{12}$: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$ = $2\sqrt{3}$

Simplify $\sqrt{27}$: $\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$ = $3\sqrt{3}$

Check if now like radicals: $2\sqrt{3}$ and $3\sqrt{3}$ both have $\sqrt{3}$ = Like radicals

Add the coefficients: $2 + 3 = 5$ = $5\sqrt{3}$

Answer: $5\sqrt{3}$

Multiple Terms with Simplification

Simplify: $\sqrt{50} - \sqrt{8} + \sqrt{18}$

Simplify $\sqrt{50}$: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ = $5\sqrt{2}$

Simplify $\sqrt{8}$: $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$ = $2\sqrt{2}$

Simplify $\sqrt{18}$: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ = $3\sqrt{2}$

Combine like radicals: $5\sqrt{2} - 2\sqrt{2} + 3\sqrt{2} = (5 - 2 + 3)\sqrt{2}$ = $6\sqrt{2}$

Answer: $6\sqrt{2}$

Unlike Radicals Cannot Combine

Simplify: $3\sqrt{5} + 2\sqrt{7}$

Check if radicals are like: $\sqrt{5}$ and $\sqrt{7}$ have different radicands = Unlike radicals

Can we simplify either radical?: 5 and 7 are both prime, no perfect square factors = Cannot simplify

Conclusion: Unlike radicals cannot be combined = Leave as is

Answer: $3\sqrt{5} + 2\sqrt{7}$ (cannot be simplified further)

Mixed: Some Combine, Some Don't

Simplify: $2\sqrt{3} + 5\sqrt{2} + 4\sqrt{3} - \sqrt{2}$

Group like radicals: $(2\sqrt{3} + 4\sqrt{3}) + (5\sqrt{2} - \sqrt{2})$ = Grouped

Combine $\sqrt{3}$ terms: $2\sqrt{3} + 4\sqrt{3} = 6\sqrt{3}$ = $6\sqrt{3}$

Combine $\sqrt{2}$ terms: $5\sqrt{2} - 1\sqrt{2} = 4\sqrt{2}$ = $4\sqrt{2}$

Write final answer: $6\sqrt{3} + 4\sqrt{2}$ = Cannot combine further

Answer: $6\sqrt{3} + 4\sqrt{2}$

Mistake: Adding radicands: $\sqrt{3} + \sqrt{5} = \sqrt{8}$

Why: You cannot add numbers under different radicals. This is like saying 1 apple + 1 orange = 2 apporanges!

Correct: $\sqrt{3} + \sqrt{5}$ cannot be simplified. They are unlike radicals.

Mistake: Forgetting to simplify first: $\sqrt{8} + \sqrt{2} = \sqrt{8} + \sqrt{2}$

Why: If you simplify $\sqrt{8} = 2\sqrt{2}$ first, you can combine: $2\sqrt{2} + \sqrt{2} = 3\sqrt{2}$

Correct: Always simplify radicals first, then check if they become like radicals.

Mistake: Combining coefficients AND radicands: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{10}$

Why: Only the coefficients change. The radicand stays the same.

Correct: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$

Mistake: Ignoring the coefficient of 1: $\sqrt{7} + 2\sqrt{7} = 2\sqrt{7}$

Why: $\sqrt{7}$ means $1\sqrt{7}$. So $1 + 2 = 3$, not just $2$.

Correct: $\sqrt{7} + 2\sqrt{7} = 1\sqrt{7} + 2\sqrt{7} = 3\sqrt{7}$

Perimeter of a Triangle

When finding perimeters of shapes with sides expressed as radicals, you need to add them.

A triangle has sides $\sqrt{12}$ m, $\sqrt{27}$ m, and $\sqrt{48}$ m. The perimeter is $2\sqrt{3} + 3\sqrt{3} + 4\sqrt{3} = 9\sqrt{3}$ meters.

Construction and Architecture

Architects often work with diagonal measurements that involve square roots.

Two roof sections have lengths $5\sqrt{3}$ meters and $2\sqrt{3}$ meters. Combined length: $7\sqrt{3}$ meters.

Q: Can I add $\sqrt{4}$ and $\sqrt{9}$?

A: Yes, but simplify first! $\sqrt{4} = 2$ and $\sqrt{9} = 3$, so $\sqrt{4} + \sqrt{9} = 2 + 3 = 5$. Note this is NOT $\sqrt{13}$!

Q: How do I know if radicals are like radicals?

A: Check two things: (1) same index (square root, cube root, etc.) and (2) same radicand (number under the radical). $\sqrt{7}$ and $\sqrt{7}$ are like. $\sqrt{7}$ and $\sqrt{11}$ are not.

Q: What if there's no coefficient written?

A: No coefficient means the coefficient is 1. So $\sqrt{5} = 1\sqrt{5}$. When adding $\sqrt{5} + 3\sqrt{5}$, you get $1\sqrt{5} + 3\sqrt{5} = 4\sqrt{5}$.

Q: Can I ever combine unlike radicals?

A: Only if they simplify to like radicals. For example, $\sqrt{12} + \sqrt{27}$ looks unlike, but simplifies to $2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$.