Simplifying Radicals Practice Problems

The full extraction technique is explained in the Simplifying Radicals lesson. This practice set turns that technique into speed, asking for the one checkable number a simplified radical always produces — its coefficient, or the exponent left on a variable.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	Extract a coefficient from a square root	, coefficient
Medium	Extract a coefficient from a cube root, or an exponent from a variable	, exponent
Hard	Bigger coefficients, a cube-root variable exponent, or a genuine decimal approximation

Easy extracts the coefficient when simplifying a square root, using radicands built so the extraction always lands on a whole number.

Medium alternates between a cube-root coefficient extraction and a square-root variable problem, asking for the exponent left outside the radical.

Hard rotates through three shapes: a bigger square-root coefficient extraction, a cube-root variable exponent, and a genuine decimal approximation of a cube root that isn’t a perfect cube.

Worked examples

One example at each difficulty level, solved the way the practice set expects.

Easy. Simplify .

Find the largest perfect-square factor: , so . The coefficient is .

Medium. Simplify and .

, so . For , take out as many pairs as possible: , giving .

Hard. Simplify and estimate .

, so . For , and , so it’s about .

Using the practice problems

Dial in a difficulty and a question count, and a new set appears right on the page. Whole-number answers go in as-is; the decimal-approximation problems accept a small rounding tolerance.

Check Answers checks the whole set immediately and marks the ones to look at again
Show Answers reveals the complete solution set, good for marking a printed attempt
Print Worksheet creates a printable page on the spot, good for homework or handouts

The rules these practice problems drill

Rule	Statement
Product Rule for Radicals
Extracting a perfect square
Extracting a variable exponent

See the Simplifying Radicals lesson for the full reasoning and how to find the largest perfect-power factor by hand.

Common mistakes to watch for

Extracting a smaller perfect square than the largest available. The answer isn’t wrong, but it isn’t fully simplified either — keep extracting until nothing perfect-square is left under the root.

Splitting a variable exponent incorrectly. , not followed by extracting the wrong piece — always take the largest exponent that’s a multiple of the index.

Mixing up perfect squares with perfect cubes. is a perfect square but not a perfect cube; check the list matching the actual index before extracting.

Treating a sum under the radical like a product. — the Product Rule for Radicals only applies to a product or quotient under the root.

Where to go next

Once extraction feels automatic, the Rationalizing Denominators Practice Problems is the natural next step, since a simplified radical is exactly what a rationalizing multiplier is built from. For the full technique and every extraction example, see the Simplifying Radicals lesson.

Frequently Asked Questions

Why does every problem ask for just the coefficient, not the full simplified radical?+

A fully simplified radical like 6√2 is an expression with two parts, which can't be automatically checked as a single number. Asking for the coefficient alone still requires finding the largest perfect-power factor correctly — the same skill, with one checkable number at the end.

How do I find the coefficient without doing the whole radical?+

Divide the radicand by perfect squares (4, 9, 16, 25, ...) until you find the largest one that divides evenly, then take its square root — that square root is the coefficient. The same idea applies to cube roots using perfect cubes instead.

What does 'the exponent on x' mean in the variable problems?+

When simplifying √(x^k), the exponent that ends up outside the radical is the largest whole number of pairs of x's that can be pulled out, which is k divided by 2 and rounded down. The same idea applies to a cube root, dividing by 3 instead of 2.

Why is one Hard subtype a decimal approximation instead of an exact answer?+

Not every radicand is a perfect cube, and a genuinely unsimplifiable cube root — like the cube root of 20 — is still a real number that can be approximated. That subtype checks the numeric value directly, accepting a small rounding tolerance.

Do I need Rationalizing Denominators before this practice set?+

No — simplifying a radical and rationalizing a denominator are separate skills, and this practice set only covers the first one. See the Rationalizing Denominators Practice Problems for the second.

Is unlimited practice really unlimited?+

There's no ceiling on this. Unlimited free sets, generated instantly, no account needed.

√Simplifying Radicals Practice Problems