Simplification Calculator

A radical is in simplest form when three things are true: no perfect-square factor is left under the root, no fraction is under the root, and no radical is left in a denominator.

The engine is the product rule $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$. To reduce $\sqrt{n}$, split off the largest perfect square inside it and take its root outside.

$\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$

When the number is awkward, prime factorise and pull out pairs: $\sqrt{200} = \sqrt{2^3 \cdot 5^2}$, so one pair of 2s and one pair of 5s escape as $2 \times 5 = 10$, leaving $10\sqrt{2}$.

Some radicals cannot be reduced at all. $\sqrt{15} = \sqrt{3 \cdot 5}$ has no repeated prime, so it is already simplest — "cannot be reduced" is a complete answer, not a failure.

A numerical check catches most slips: $\sqrt{200} \approx 14.142$ and $10\sqrt{2} \approx 14.142$, so that rewrite preserved the value.

Variables follow the same pair rule. For $x \ge 0$, $\sqrt{x^2} = x$, $\sqrt{x^4} = x^2$ — halve the exponent whenever it is even. An odd exponent leaves one factor behind:

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

Deal with the number and each variable separately, then multiply the escaped parts together.

Multiplying radicals goes the other way: combine first, then reduce. $\sqrt{6} \cdot \sqrt{10} = \sqrt{60}$, and $60 = 4 \cdot 15$, so the answer is $2\sqrt{15}$.

Rationalising removes a radical from the denominator. With a single term, multiply top and bottom by that radical, since $\sqrt{a} \cdot \sqrt{a} = a$. With a two-term denominator, multiply by the conjugate — flip the middle sign — because $(a + b)(a - b) = a^2 - b^2$ clears both radicals at once.

• Pulling out a factor that is not a square. In $\sqrt{18}$ only the 9 escapes: $3\sqrt{2}$. The 2 stays put.
• Stopping halfway. $\sqrt{72} = 2\sqrt{18}$ is true but not simplest, because 18 still contains a 9. Always recheck the leftover.
• Halving an odd exponent. $\sqrt{x^3}$ is $x\sqrt{x}$, not $x^{1.5}$ in radical form — one $x$ is left inside.
• Dropping the absolute value. $\sqrt{x^2} = |x|$ in general; you may write $x$ only when the problem states $x \ge 0$.
• Splitting a root over a sum. $\sqrt{x^2 + 9} \neq x + 3$. The product rule works for multiplication and division only.
• Multiplying only the denominator when rationalising. Both parts of the fraction must be multiplied, or you have changed its value.