Simplifying Radicals and Radical Operations

√ Simplifying Radicals

Part 1 of 5 — Square Roots & Perfect Squares

Topics in This Part

🔑 Key Concept: A radical like $\sqrt{72}$ is simplified by pulling out the largest perfect square hiding inside it. Everything in this lesson rests on spotting perfect squares — so that's where we start.

What a Square Root Means

The square root of a number $n$, written $\sqrt{n}$, is the value that multiplies by itself to give $n$:

$\sqrt{n} = a \quad\text{means}\quad a \cdot a = n \quad (a \ge 0)$

The symbol $\sqrt{\;\;}$ is the radical sign, and the number inside it is the radicand.

💡 By convention $\sqrt{\;\;}$ means the principal (non-negative) root. So $\sqrt{25} = 5$, not $-5$.

Match the Roots 🔽

Perfect Squares

A perfect square is a whole number whose square root is also a whole number. Memorizing the first dozen makes simplifying radicals fast:

$1,\; 4,\; 9,\; 16,\; 25,\; 36,\; 49,\; 64,\; 81,\; 100,\; 121,\; 144$

🔑 Key Idea: When the radicand is not a perfect square (like $\sqrt{72}$), it can't become a whole number — but it can still be simplified by factoring out a perfect square.

Concept Check 🎯

Finding the Largest Perfect-Square Factor

To simplify a radical, look for the largest perfect square that divides the radicand.

Example: $\sqrt{72}$

Factors of $72$ include $4, 9,$ and $36$. The largest perfect square is $36$:

$72 = 36 \cdot 2 \quad\Rightarrow\quad \sqrt{72} = \sqrt{36}\cdot\sqrt{2} = 6\sqrt{2}$

💡 You don't have to find the largest factor on the first try. If you pull out $\sqrt{72} = \sqrt{4}\cdot\sqrt{18} = 2\sqrt{18}$, you simply repeat the process on $\sqrt{18} = 3\sqrt{2}$, giving $2\cdot 3\sqrt{2} = 6\sqrt{2}$ — the same answer. Finding the largest square just saves a step.

Largest Perfect-Square Factor 🧮

Enter the largest perfect square that divides each number.

1) $\;48$ 2) $\;50$ 3) $\;75$

√ Simplifying Radicals

Part 2 of 5 — The Product & Quotient Rules

🔑 The Idea: Radicals split across multiplication and division. That single fact is what lets us pull perfect squares out of any radicand — numbers and variables.

The Two Core Rules

For non-negative numbers $a$ and $b$:

$\textbf{Product Rule:}\quad \sqrt{a \cdot b} = \sqrt{a}\cdot\sqrt{b}$

$\textbf{Quotient Rule:}\quad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \quad (b \ne 0)$

These read both directions — you can split a radical apart or combine two radicals into one.

Worked Example: $\sqrt{48}$

$\sqrt{48} = \sqrt{16\cdot 3} = \sqrt{16}\cdot\sqrt{3} = 4\sqrt{3}$

Worked Example: $\sqrt{\dfrac{9}{25}}$

$\sqrt{\frac{9}{25}} = \frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5}$

⚠️ The rules apply to multiplication and division only. $\sqrt{a+b} \ne \sqrt{a} + \sqrt{b}$. For example $\sqrt{9+16} = \sqrt{25} = 5$, but $\sqrt{9}+\sqrt{16} = 3+4 = 7$. Not equal!

Concept Check 🎯

Radicals with Variables

A variable is a perfect square when its exponent is even, because $\sqrt{x^{2}} = x$ (assuming $x \ge 0$).

$\sqrt{x^{2}} = x,\qquad \sqrt{x^{4}} = x^{2},\qquad \sqrt{x^{6}} = x^{3}$

For odd powers, split off one factor of $x$:

$\sqrt{x^{5}} = \sqrt{x^{4}\cdot x} = x^{2}\sqrt{x}$

Worked Example: $\sqrt{18x^{3}}$

Break the number and the variable separately:

$\sqrt{18x^{3}} = \sqrt{9\cdot 2 \cdot x^{2}\cdot x} = \sqrt{9}\cdot\sqrt{x^{2}}\cdot\sqrt{2x} = 3x\sqrt{2x}$

💡 Shortcut: Halve each even exponent to bring it outside; an odd exponent leaves one copy inside. $\sqrt{x^{7}} = x^{3}\sqrt{x}$ (half of $6$ is $3$, one $x$ stays in).

Simplify with Variables 🔽

Putting It Together

Most problems mix numbers and variables. Handle the number and each variable separately, then collect what comes out:

Worked Example: $\sqrt{50x^{4}y^{3}}$

$\sqrt{50x^{4}y^{3}} = \sqrt{25 \cdot 2 \cdot x^{4} \cdot y^{2} \cdot y} = 5x^{2}y\sqrt{2y}$

• $25 \to 5$ comes out, $2$ stays in
• $x^{4} \to x^{2}$ comes out (nothing left in)
• $y^{3} \to y$ comes out, one $y$ stays in

🔑 Everything with an even count escapes the radical; whatever has an odd count leaves exactly one factor behind.

Simplify 🧮

Write each in simplest radical form. Enter the whole-number coefficient in front of the radical (the number outside).

1) $\sqrt{75} = a\sqrt{3},\quad a = \,?$ 2) $\sqrt{32} = a\sqrt{2},\quad a = \,?$ 3) $\sqrt{12x^{2}} = ax\sqrt{3},\quad a = \,?$ (assume $x \ge 0$)

√ Simplifying Radicals

Part 3 of 5 — Adding & Subtracting Radicals

🔑 The Big Rule: You can only add or subtract radicals that are like radicals — same radicand. They behave exactly like combining like terms in algebra.

Like Radicals = Like Terms

Think of $\sqrt{2}$ as a "unit," just like $x$. You can combine matching units:

$3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2} \qquad\text{(just like } 3x + 4x = 7x\text{)}$

But you cannot combine unlike radicals:

$3\sqrt{2} + 4\sqrt{5} \quad\text{stays as}\quad 3\sqrt{2} + 4\sqrt{5}$

⚠️ Add the coefficients only — the radicand never changes. $3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2}$, not $7\sqrt{4}$.

Concept Check 🎯

Simplify First, Then Combine

Radicals that look different may become like radicals after simplifying. Always simplify each radical first.

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

Neither is simplified yet:

$\sqrt{18} = \sqrt{9\cdot 2} = 3\sqrt{2}, \qquad \sqrt{8} = \sqrt{4\cdot 2} = 2\sqrt{2}$

Now they're like radicals:

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

Worked Example: $\sqrt{75} - \sqrt{12}$

$\sqrt{75} = 5\sqrt{3}, \qquad \sqrt{12} = 2\sqrt{3} \quad\Rightarrow\quad 5\sqrt{3} - 2\sqrt{3} = 3\sqrt{3}$

💡 Two radicals that seem unlike often hide the same simplified radicand. $\sqrt{18}$ and $\sqrt{8}$ both reduce to a multiple of $\sqrt{2}$.

Simplify Then Combine 🔽

Work through $\sqrt{50} + \sqrt{8}$ step by step.

Three or More Terms

The same idea scales up. Simplify every radical, then group like radicands.

Worked Example: $2\sqrt{3} + \sqrt{12} - \sqrt{27}$

$\sqrt{12} = 2\sqrt{3}, \qquad \sqrt{27} = 3\sqrt{3}$

$2\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} = (2 + 2 - 3)\sqrt{3} = 1\sqrt{3} = \sqrt{3}$

💡 Line up coefficients and add/subtract them in order — the shared radical $\sqrt{3}$ just rides along unchanged.

Add & Subtract 🧮

Simplify completely. Enter the coefficient in front of the resulting radical.

1) $\sqrt{27} + \sqrt{12} = a\sqrt{3},\quad a = \,?$ 2) $\sqrt{45} - \sqrt{20} = a\sqrt{5},\quad a = \,?$

√ Simplifying Radicals

Part 4 of 5 — Multiplying Radicals

🔑 The Idea: Multiply the outsides together and the insides together, using the product rule $\sqrt{a}\cdot\sqrt{b} = \sqrt{ab}$ — then simplify the result.

Multiplying Single Radicals

Multiply coefficients with coefficients, radicands with radicands, then simplify:

$\left(a\sqrt{m}\right)\left(b\sqrt{n}\right) = ab\,\sqrt{mn}$

Worked Example: $\sqrt{6}\cdot\sqrt{2}$

$\sqrt{6}\cdot\sqrt{2} = \sqrt{12} = \sqrt{4\cdot 3} = 2\sqrt{3}$

Worked Example: $\left(3\sqrt{5}\right)\left(2\sqrt{10}\right)$

$= (3\cdot 2)\,\sqrt{5\cdot 10} = 6\sqrt{50} = 6\cdot 5\sqrt{2} = 30\sqrt{2}$

A Radical Times Itself

$\sqrt{7}\cdot\sqrt{7} = \sqrt{49} = 7 \qquad\text{In general: } \left(\sqrt{a}\right)^2 = a$

💡 A square root times itself removes the radical entirely. This is the engine behind rationalizing denominators in Part 5.

Concept Check 🎯

Distributing & FOIL with Radicals

Radicals follow the distributive property and FOIL just like polynomials.

Distribute: $\sqrt{2}\left(\sqrt{6} + \sqrt{10}\right)$

$= \sqrt{2}\cdot\sqrt{6} + \sqrt{2}\cdot\sqrt{10} = \sqrt{12} + \sqrt{20} = 2\sqrt{3} + 2\sqrt{5}$

FOIL: $\left(\sqrt{3} + 2\right)\left(\sqrt{3} + 5\right)$

$\underbrace{\sqrt{3}\cdot\sqrt{3}}_{3} + \underbrace{5\sqrt{3}}_{\text{O}} + \underbrace{2\sqrt{3}}_{\text{I}} + \underbrace{10}_{\text{L}} = 3 + 7\sqrt{3} + 10 = 13 + 7\sqrt{3}$

💡 The First product $\sqrt{3}\cdot\sqrt{3} = 3$ loses its radical, while the Outer/Inner terms ($5\sqrt{3}$ and $2\sqrt{3}$) are like radicals you combine.

FOIL the Product 🔽

Expand $\left(\sqrt{5} + 1\right)\left(\sqrt{5} - 3\right)$ piece by piece.

A Special Product to Remember

Multiplying conjugates — a sum times a difference — wipes out the radicals through difference of squares:

$\left(\sqrt{a} + \sqrt{b}\right)\left(\sqrt{a} - \sqrt{b}\right) = \left(\sqrt{a}\right)^2 - \left(\sqrt{b}\right)^2 = a - b$

Worked Example

$\left(\sqrt{7} + \sqrt{3}\right)\left(\sqrt{7} - \sqrt{3}\right) = 7 - 3 = 4$

🔑 Keep this in your pocket — it's the exact trick that rationalizes binomial denominators in Part 5.

Multiply 🧮

Simplify each product completely.

1) $\sqrt{6}\cdot\sqrt{3} = a\sqrt{2},\quad a = \,?$ 2) $\left(4\sqrt{2}\right)\left(3\sqrt{2}\right) = \,?$ (a whole number) 3) $\left(\sqrt{5}\right)^2 = \,?$ (a whole number)

√ Simplifying Radicals

Part 5 of 5 — Rationalizing Denominators & Mastery Check

🔑 The Rule of Form: A radical expression isn't considered fully simplified while a radical sits in the denominator. Rationalizing clears it out.

Rationalizing a Single-Term Denominator

Multiply the top and bottom by the radical in the denominator. Since $\sqrt{a}\cdot\sqrt{a} = a$, the bottom becomes rational.

Worked Example: $\dfrac{1}{\sqrt{3}}$

$\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{9}} = \frac{\sqrt{3}}{3}$

Worked Example: $\dfrac{6}{\sqrt{2}}$

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

💡 Multiplying by $\dfrac{\sqrt{a}}{\sqrt{a}}$ is multiplying by $1$, so the value never changes — only the form does.

Concept Check 🎯

Rationalizing with a Conjugate

When the denominator is a binomial like $\sqrt{5} + \sqrt{2}$, multiply by its conjugate — the same terms with the opposite middle sign. This uses the difference of squares to erase both radicals:

$\left(\sqrt{a} + \sqrt{b}\right)\left(\sqrt{a} - \sqrt{b}\right) = a - b$

Worked Example: $\dfrac{1}{\sqrt{5} + \sqrt{2}}$

Multiply top and bottom by the conjugate $\sqrt{5} - \sqrt{2}$:

$\frac{1}{\sqrt{5} + \sqrt{2}}\cdot\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{\sqrt{5} - \sqrt{2}}{5 - 2} = \frac{\sqrt{5} - \sqrt{2}}{3}$

⚠️ The conjugate of $\sqrt{5} + \sqrt{2}$ is $\sqrt{5} - \sqrt{2}$ — only the middle sign flips. Don't change the order of the terms.

Conjugates & Forms 🔽

Quick Reference

⚠️ Remember: $\sqrt{a+b} \ne \sqrt{a} + \sqrt{b}$, and you can only combine radicals with the same radicand.

Mixed Practice 🎯

You've Built the Full Toolkit

You can now move fluently between every radical skill:

💡 On a test, always ask two questions at the end: Is there a perfect square still hiding inside? and Is there a radical still in the denominator? If both answers are "no," you're done.

Exit Quiz ✅

Answer all three to finish the lesson.