Adding and Subtracting Radicals

√ Adding and Subtracting Radicals

Part 1 of 5 — Radicals and "Like" Terms

Topics in This Part

🔑 Key Concept: You can only add or subtract radicals that are alike — same index and same radicand. It works exactly like combining $3x + 5x = 8x$: you add the numbers in front and keep the radical the same.

Anatomy of a Radical

Every radical expression has three parts:

$\underset{\text{coefficient}}{3}\,\overset{\text{index}}{\sqrt[2]{\underset{\text{radicand}}{7}}}$

When no number is written in front, the coefficient is $1$: $\sqrt{2} = 1\sqrt{2}$.

When no small index is written, it's a square root (index $2$): $\sqrt{2} = \sqrt[2]{2}$.

💡 Think of $\sqrt{2}$ as a single "unit," like the variable $x$. Then $5\sqrt{2}$ is just "5 of those units."

Name the Parts 🔽

Look at the radical $7\sqrt{3}$ and identify each piece.

What Makes Radicals "Like"

Two radicals are like radicals when they have the same index and the same radicand. The coefficient out front does not have to match.

🔑 The Rule: Same index and same radicand $\Rightarrow$ like radicals $\Rightarrow$ you can combine them. Otherwise, leave them separate.

Concept Check 🎯

Sort Them 🔽

For each pair, decide whether they are like or unlike radicals.

Why This Matters

You can only add or subtract like radicals. Trying to combine unlike radicals is the #1 error in this whole topic.

⚠️ Watch out: $\sqrt{2} + \sqrt{3} \ne \sqrt{5}$. You cannot add the radicands! ($\sqrt{2}+\sqrt{3} \approx 1.41 + 1.73 = 3.15$, but $\sqrt{5} \approx 2.24$.) They are unlike, so $\sqrt{2}+\sqrt{3}$ stays exactly as it is.

Now that you can spot like radicals, Part 2 shows you how to actually combine them.

√ Adding and Subtracting Radicals

Part 2 of 5 — Combining Like Radicals

🔑 The Idea: To combine like radicals, add (or subtract) the coefficients and keep the radical the same. The radicand never changes.

The Method

Treat the radical like a variable. Compare:

$3x + 5x = 8x \qquad\Longleftrightarrow\qquad 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$

You add the numbers in front; the "$\sqrt{2}$" rides along unchanged, just like the "$x$."

Worked Example: $3\sqrt{2} + 5\sqrt{2}$

Both terms are $\sqrt{2}$, so add the coefficients $3 + 5 = 8$:

$3\sqrt{2} + 5\sqrt{2} = (3+5)\sqrt{2} = 8\sqrt{2}$

Worked Example: $9\sqrt{7} - 4\sqrt{7}$

Subtract the coefficients $9 - 4 = 5$:

$9\sqrt{7} - 4\sqrt{7} = (9-4)\sqrt{7} = 5\sqrt{7}$

⚠️ Never add the radicands. $3\sqrt{2} + 5\sqrt{2}$ is $8\sqrt{2}$, not $8\sqrt{4}$ and definitely not $\sqrt{10}$.

Fill in the Blanks 🔽

Combine $6\sqrt{3} + 2\sqrt{3}$, paralleling $6x + 2x$.

The Invisible Coefficient of 1

When a radical has no number in front, its coefficient is $1$:

$\sqrt{6} + \sqrt{6} = 1\sqrt{6} + 1\sqrt{6} = 2\sqrt{6}$

$5\sqrt{3} - \sqrt{3} = 5\sqrt{3} - 1\sqrt{3} = 4\sqrt{3}$

💡 Forgetting the hidden $1$ is a classic slip. $5\sqrt{3} - \sqrt{3}$ is $4\sqrt{3}$, not $5\sqrt{3}$.

Concept Check 🎯

When You Can't Combine

If the radicals are unlike, you simply leave the expression as it is — it's already simplified.

$2\sqrt{3} + 5\sqrt{7} \quad\text{stays}\quad 2\sqrt{3} + 5\sqrt{7}$

There is nothing to add: $\sqrt{3}$ and $\sqrt{7}$ are different "units."

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

Group only the like terms ($\sqrt{2}$ with $\sqrt{2}$):

$\underbrace{4\sqrt{2} - \sqrt{2}}_{=\,3\sqrt{2}} + 3\sqrt{5} = 3\sqrt{2} + 3\sqrt{5}$

The $3\sqrt{5}$ has no partner, so it stays. Final answer: $3\sqrt{2} + 3\sqrt{5}$.

💡 Strategy: it helps to color-code or underline like radicals before combining, just like collecting like terms in a polynomial.

Combine the Coefficients 🧮

Each answer has the form (coefficient)$\sqrt{\text{radicand}}$. Enter just the coefficient (the number in front).

1) $7\sqrt{2} + 2\sqrt{2} = \;?\,\sqrt{2}$ 2) $10\sqrt{6} - 3\sqrt{6} = \;?\,\sqrt{6}$ 3) $\sqrt{11} + \sqrt{11} + \sqrt{11} = \;?\,\sqrt{11}$

√ Adding and Subtracting Radicals

Part 3 of 5 — Simplify First, Then Combine

🔑 The Big Insight: Radicals that look unlike are often secretly like — once you simplify them. Always simplify each radical before deciding whether terms combine.

Simplifying a Radical (Quick Review)

To simplify $\sqrt{n}$, pull out the largest perfect-square factor:

$\sqrt{a \cdot b} = \sqrt{a}\cdot\sqrt{b}$

💡 Perfect squares to memorize: $4, 9, 16, 25, 36, 49, 64, 81, 100$. Look for the largest one that divides your radicand.

Simplify Each Radical 🧮

Each simplifies to (coefficient)$\sqrt{2}$. Enter just the coefficient out front.

1) $\sqrt{8} = \;?\,\sqrt{2}$ 2) $\sqrt{32} = \;?\,\sqrt{2}$ 3) $\sqrt{72} = \;?\,\sqrt{2}$

The Hidden-Like-Radicals Trick

Worked Example: $\sqrt{8} + \sqrt{18}$

At first glance these look unlike. Simplify each:

$\sqrt{8} = 2\sqrt{2} \qquad \sqrt{18} = 3\sqrt{2}$

Now they're both $\sqrt{2}$ — like radicals! Combine:

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

Worked Example: $\sqrt{50} - \sqrt{8}$

$\sqrt{50} = 5\sqrt{2}, \qquad \sqrt{8} = 2\sqrt{2}$

$5\sqrt{2} - 2\sqrt{2} = 3\sqrt{2}$

🔑 Golden Rule: Simplify every radical first. Only then check for like radicals and combine.

Concept Check 🎯

Sometimes They Stay Unlike

Simplifying doesn't always make radicals match. If the simplified radicands differ, leave the terms separate.

Worked Example: $\sqrt{12} + \sqrt{50}$

$\sqrt{12} = 2\sqrt{3}, \qquad \sqrt{50} = 5\sqrt{2}$

Now we have $2\sqrt{3} + 5\sqrt{2}$ — radicands $3$ and $2$ are different, so this is the final simplified answer. You can't combine them.

⚠️ Don't force a combination. $2\sqrt{3} + 5\sqrt{2}$ is fully simplified, even though it has two terms.

Simplify, Then Combine 🧮

Each result is (coefficient)$\sqrt{r}$. Enter the coefficient out front of the final answer.

1) $\sqrt{18} + \sqrt{2} = \;?\,\sqrt{2}$ 2) $\sqrt{75} - \sqrt{12} = \;?\,\sqrt{3}$ 3) $\sqrt{32} + \sqrt{8} = \;?\,\sqrt{2}$

√ Adding and Subtracting Radicals

Part 4 of 5 — Multi-Term Problems & Applications

🔑 Now we level up: three or more terms, two different radicands at once, coefficients out front of an already-simplifiable radical, and a real geometry application.

Multiple Radicands at Once

When an expression mixes two (or more) radicands, simplify everything, then group like with like.

Worked Example: $\sqrt{8} + \sqrt{27} + \sqrt{2} + \sqrt{12}$

Simplify each term:

$\sqrt{8}=2\sqrt{2},\quad \sqrt{27}=3\sqrt{3},\quad \sqrt{2}=1\sqrt{2},\quad \sqrt{12}=2\sqrt{3}$

Group the $\sqrt{2}$'s and the $\sqrt{3}$'s separately:

$\underbrace{(2\sqrt{2}+\sqrt{2})}_{=\,3\sqrt{2}} + \underbrace{(3\sqrt{3}+2\sqrt{3})}_{=\,5\sqrt{3}} = 3\sqrt{2} + 5\sqrt{3}$

The two groups have different radicands, so the answer keeps both terms: $3\sqrt{2} + 5\sqrt{3}$.

Concept Check 🎯

Coefficients Times Radicals

If a term already has a coefficient out front, multiply it into the simplified radical.

Worked Example: $2\sqrt{18} - 5\sqrt{2}$

First simplify $\sqrt{18} = 3\sqrt{2}$, then multiply by the outside $2$:

$2\sqrt{18} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}$

Now combine: $6\sqrt{2} - 5\sqrt{2} = 1\sqrt{2} = \sqrt{2}$.

Worked Example: $3\sqrt{20} + \sqrt{45}$

$3\sqrt{20} = 3\cdot 2\sqrt{5} = 6\sqrt{5}, \qquad \sqrt{45} = 3\sqrt{5}$

$6\sqrt{5} + 3\sqrt{5} = 9\sqrt{5}$

⚠️ Multiply the outside coefficient by the number that comes out of the radical — not by the radicand. $2\sqrt{18} = 2(3\sqrt2) = 6\sqrt2$, not $6\sqrt{18}$.

Walk Through It 🔽

Simplify $3\sqrt{20} + \sqrt{45}$ one step at a time.

Geometry Application: Perimeter

Radicals show up whenever you add up lengths involving square roots — like a perimeter.

Worked Example

A rectangle has length $\sqrt{50}$ cm and width $\sqrt{8}$ cm. Find its perimeter, simplified.

Perimeter $= 2(\text{length}) + 2(\text{width})$:

$P = 2\sqrt{50} + 2\sqrt{8}$

Simplify each radical: $\sqrt{50} = 5\sqrt{2}$ and $\sqrt{8} = 2\sqrt{2}$.

$P = 2(5\sqrt{2}) + 2(2\sqrt{2}) = 10\sqrt{2} + 4\sqrt{2} = 14\sqrt{2}\text{ cm}$

💡 Leaving the answer as $14\sqrt{2}$ is exact. A decimal ($\approx 19.8$ cm) is only an approximation.

Multi-Step Practice 🧮

Each answer is (coefficient)$\sqrt{r}$. Enter just the coefficient out front of the simplified answer.

1) $2\sqrt{12} + 3\sqrt{3} = \;?\,\sqrt{3}$ 2) $\sqrt{8} + \sqrt{18} + \sqrt{2} = \;?\,\sqrt{2}$ 3) Perimeter of a square with side $\sqrt{27}$: $4\sqrt{27} = \;?\,\sqrt{3}$

√ Adding and Subtracting Radicals

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) recognize like radicals, (2) combine them by adding coefficients, (3) simplify radicals to reveal hidden like terms, and (4) handle multi-term problems and applications. Let's put it all together.

Quick Reference

⚠️ Top mistakes to avoid:

• Adding the radicands: $\sqrt{2}+\sqrt{3} \ne \sqrt{5}$.
• Forgetting to simplify first: $\sqrt{8}+\sqrt{2}$ really is $3\sqrt{2}$.
• Dropping the hidden $1$: $5\sqrt{3}-\sqrt{3} = 4\sqrt{3}$.

Mixed Practice 🎯

Combine or Not? 🔽

For each expression, choose the fully simplified result.

Final Drill 🧮

Enter the coefficient out front of each fully simplified answer.

1) $\sqrt{20} + \sqrt{45} = \;?\,\sqrt{5}$ 2) $4\sqrt{2} + \sqrt{8} - \sqrt{2} = \;?\,\sqrt{2}$ 3) $\sqrt{98} - \sqrt{32} = \;?\,\sqrt{2}$

Exit Quiz ✅

Answer all three to finish the lesson.