Simplifying and Operating on Radicals

A radicand often hides a perfect square. Pull that perfect square out of the root and you get a smaller, standard-form radical — and once two radicals share the same index and radicand, you can add or subtract them like ordinary like terms.

By the end you'll be able to factor a radicand into a perfect square times a remainder, extract the perfect square, and combine like radicals while leaving unlike ones alone.

Predict: what is the simplified form of \(\sqrt{50}\)? Factor out the perfect square below and check.

Drag the radicand slider to reshape the bar — the perfect-square part splits off and its root walks outside the radical as a coefficient, leaving the remainder behind under the root. Then toggle the second term below to see like radicals combine while unlike ones can't.

\(\sqrt{50} = \sqrt{25}\cdot\sqrt{2} = \) 5\(\sqrt{2}\)

Factor out the perfect square, then combine like radicals
perfect-square factor → extracted coefficient remainder → stays under the root

Radicals reach standard form with two moves: pull perfect powers out from under the root, and clear any radical out of a denominator — then only radicals that match exactly can be added or subtracted.

Procedural

To simplify a radical, split the radicand into a perfect square times whatever's left, then apply the product rule in reverse: \(\sqrt{n} = \sqrt{p \cdot r} = \sqrt{p}\cdot\sqrt{r}\). If \(p\) is a perfect square, \(\sqrt{p}\) is a whole number that walks outside the root, leaving \(r\) behind. Once every radical you're adding is in this standard form, combine the ones that match:

  • Like radicals (same index, same radicand) add or subtract their coefficients, the same way \(3x + 5x = 8x\): \(5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}\).
  • Unlike radicals never merge — \(\sqrt{2} + \sqrt{3}\) stays exactly as written, just as \(x + y\) doesn't collapse into one term.
  • Radicals that look unlike can become like after simplifying: \(\sqrt{8}=2\sqrt{2}\), which now matches \(\sqrt{2}\).
Formal

The product and quotient rules split a root over multiplication and division only — never over addition: \(\sqrt{a+b}\ne\sqrt{a}+\sqrt{b}\). Standard form also forbids a radical in a denominator. For a single root, multiply by that root over itself: \(\dfrac{3}{\sqrt{5}} = \dfrac{3}{\sqrt5}\cdot\dfrac{\sqrt5}{\sqrt5} = \dfrac{3\sqrt5}{5}\). For a binomial denominator, multiply by its conjugate (flip the middle sign) to exploit \((a-b)(a+b)=a^2-b^2\), which clears the root from the bottom entirely.

Applied

A carpenter computing a diagonal brace length with the Pythagorean theorem, or a physicist working out a period formula, routinely lands on an answer like \(\sqrt{72}\) inches. Simplifying it to \(6\sqrt{2}\) makes it easier to compare, combine with other measurements, or round for a cut list — the value is identical, just written in the standard form everyone expects.

Worked example

Simplify \(\sqrt{48} + \sqrt{27}\). Factor each radicand: \(48 = 16\times 3\), so \(\sqrt{48}=4\sqrt{3}\); \(27 = 9\times 3\), so \(\sqrt{27}=3\sqrt{3}\). Both are now radical-3 terms — like radicals — so add their coefficients: \(4\sqrt{3}+3\sqrt{3} = \) \(7\sqrt{3}\).

Your turn

Simplify \(\sqrt{80} - \sqrt{45}\). \(80 = 16\times 5\), so \(\sqrt{80}=4\sqrt{5}\). \(45 = 9\times 5\), so \(\sqrt{45}=3\sqrt{5}\). Both are radical-5 terms, so subtract the coefficients: \(4\sqrt{5} - 3\sqrt{5} = \) ____

Reveal the answer

\(4\sqrt{5} - 3\sqrt{5} = \) \(\sqrt{5}\) — one radical-5 left over. Set the radicand slider above to 20 or 45 in the diagram and check that each extracts the same radicand of 5.

More info — why unlike radicals stay separate

Think of \(\sqrt{3}\) as its own kind of unit, the way \(x\) is in algebra: you can collect \(3\sqrt{3}+5\sqrt{3}\) into \(8\sqrt{3}\) because they're multiples of the same unit, but \(\sqrt{2}\) is a genuinely different unit — combining \(\sqrt{2}+\sqrt{3}\) would be like trying to simplify \(x+y\) into a single variable. That's also why matching the index matters: \(\sqrt{2}\) and \(\sqrt[3]{2}\) aren't like radicals even though the radicand is the same, since one asks "what squares to 2" and the other "what cubes to 2" — different questions with different answers. See the Purplemath link in Dive deeper for more practice spotting like radicals.

Check your understanding

Question 1 of 4

Simplify \(\sqrt{98}\).

Question 2 of 4

Simplify \(6\sqrt{3} - 2\sqrt{3} + \sqrt{12}\).

Question 3 of 4

Which rational-exponent expression is the same quantity as \(\sqrt[3]{x^2}\)?

Question 4 of 4

Why can't \(\sqrt{7} + \sqrt{11}\) be written as a single radical term?

Recap

  • Simplify a radical by factoring out the largest perfect power: \(\sqrt{n}=\sqrt{p}\cdot\sqrt{r}\), extracting \(\sqrt{p}\) as a coefficient.
  • Like radicals (same index, same radicand) add or subtract their coefficients, just like like terms: \(5\sqrt3+2\sqrt3=7\sqrt3\).
  • Unlike radicals never combine — simplify first, since radicals that look different can become like after factoring (\(\sqrt8=2\sqrt2\)).
  • The product/quotient rules split a root over multiplication and division only, never addition: \(\sqrt{a+b}\ne\sqrt a+\sqrt b\).
  • Standard form has no radical in a denominator — multiply by the root itself (single term) or by the conjugate (binomial) to clear it.

Dive deeper

Sources

  • Simplifying and Operating on Radicals