Radicals

Every technique in this lesson traces back to one fact proven in Rational Exponents: a radical is a fractional exponent, . That single equivalence is why radicals obey the same product and quotient structure as exponents, and it’s what makes every rule below provable instead of just memorizable.

What Is a Radical?

A radical expression has three parts: the radical sign , the radicand underneath it (the number or expression being rooted), and the index, a small number in the sign’s notch that says which root to take. has radicand and index , and asks “what number, cubed, gives 8?” — the answer is , since .

When no index is written, it defaults to , the square root: means . A radical always returns the principal (nonnegative) root, matching the same convention used for rational exponents.

A simplifying convention. As in the previous lesson, assume every variable here is positive. That means exactly, with no absolute-value bars to track.

Why the Radical of a Power Returns the Base

This isn’t a separate rule to memorize — it falls directly out of the Power Rule for exponents. Since :

Raising the th root of back to the th power always returns — root and power are inverse operations, the same way addition and subtraction are.

Properties of Radicals

The Product and Quotient Rules are exactly how every simplification technique in this lesson works — everything below is one of those two rules applied with a specific goal.

Simplifying Radicals

To simplify , factor the radicand looking for the largest perfect th-power factor, then split it off with the Product Rule:

Stopping at a smaller factor still works arithmetically but leaves a second simplification step behind — is true, but still has a perfect-square factor left inside it. Always check the radicand one more time after pulling a factor out.

Example 1: Simplifying Radicals

Combining Like Radicals

Just as only like terms combine in a polynomial, only like radicals — the same index and the same radicand — combine, by adding or subtracting their coefficients:

cannot be combined any further; the radicands don’t match. Sometimes two radicals that look unlike become like ones once simplified — — so always simplify first, then check for like radicals.

Example 2: Combining and Multiplying Radicals

Rationalizing Denominators

A denominator with a single radical term clears by multiplying top and bottom by that same radical:

A denominator with a radical binomial, like , needs its conjugate instead — multiplying by the same radical alone won’t clear a sum. The conjugate of is , and their product is always a clean difference of squares, , with no radical left at all.

Example 3: Rationalizing Denominators

Worked Example A: Simplifying and Combining

Simplify .

Step 1 — simplify each radical separately:

Step 2 — combine, since every term is now a like radical:

Nothing can be combined until every radical is simplified first — three unlike-looking radicals turned out to be the same radical in disguise.

Worked Example B: Rationalizing with a Conjugate

Simplify .

Step 1 — multiply by the conjugate of the denominator:

Step 2 — the denominator becomes a difference of squares:

Step 3 — distribute the numerator:

Since the denominator simplified all the way to , the final answer is just .

Worked Example C: Radicals with Variables

Simplify .

Step 1 — factor out the largest perfect-square factor:

Step 2 — apply the Product Rule:

Common Mistakes to Avoid

• Stopping at a partial factorization. is technically true but incomplete — still has a perfect-square factor. Always use the largest perfect-power factor, or re-check what’s left.
• Adding radicals with different radicands or indices. stays exactly as it is; it never becomes .
• Multiplying radicals but forgetting the coefficients. , not — the coefficients multiply too.
• Assuming a radical distributes over a sum. . Test it: , but .
• Leaving a radical in the denominator. A radical answer isn’t fully simplified until the denominator is rationalized.
• Getting the conjugate’s sign backward. The conjugate of is — only the middle sign flips, not both terms.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Simplify .

Problem 2. Simplify .

Problem 3. Simplify .

Problem 4. Simplify .

Problem 5. Simplify .

Problem 6. Rationalize .

Problem 7. Rationalize .

Problem 8. Simplify .

Problem 9. Evaluate .

Problem 10. Simplify .

Quick Reference

Solving radical equations comes next, and the conjugate trick used here to rationalize a denominator reappears almost unchanged in Complex Numbers, just clearing an imaginary denominator instead of a radical one. If Rational Exponents or Integer Exponents would help as a refresher, they’re both there. For more practice, try the Radical Practice Problems; for everything else, the Algebra lessons and the site’s calculators are worth a look too.