What is a radical in math?

A radical is the symbol used to represent a root, such as a square root or cube root. It includes the radical sign √, an index that shows which root is taken (2 for square root, 3 for cube root, and so on), and the radicand, which is the number or expression inside.

What is a radical expression?

A radical expression is any expression that contains a radical sign with a number or variable inside, such as √18, 4√x, or ∛(x² + 1). Students learn to simplify these, combine like radicals, and sometimes rewrite them using rational exponents.

How do you simplify a radical?

To simplify a radical, factor the radicand to find perfect powers that match the index, such as perfect squares for square roots or perfect cubes for cube roots. Take those perfect powers out of the radical, leave any remaining factor inside, and combine with any coefficients outside the radical.

Why do radical equations sometimes have extraneous solutions?

When solving a radical equation, you often square both sides to remove the radical. Squaring can introduce extra solutions that do not work in the original equation, so each solution must be checked by substituting it back into the original radical equation.

What Is a Radical? Definition and Examples

Radical meaning for students

A radical is a way to ask, "What number makes this work if I multiply it by itself some number of times?" For example, √25 asks, "What number times itself is 25?" The answer is 5. The number under the radical sign is called the radicand, like the 25 in √25.

Worked examples with radicals

Example 1 (simpler): Simplifying a square root

Problem: Simplify √72.

Step 1: Factor 72 into a product that includes a perfect square. A perfect square is a number like 4, 9, 16, 25, or 36 that is a whole number times itself. One way is 72 = 36 × 2.

Step 2: Use the rule √(a × b) = √a × √b. So √72 = √36 × √2.

Step 3: Simplify the perfect square root. √36 = 6, so √72 = 6√2.

Answer: √72 = 6√2.

Example 2 (more advanced): Solving a radical equation and checking for extraneous solutions

Problem: Solve √(x + 3) = x − 1.

Step 1: Note the domain from the radical. The radicand x + 3 must be at least 0, so x + 3 ≥ 0 and x ≥ −3.

Step 2: Isolate the radical (it already is) and square both sides to remove the square root. (√(x + 3))² = (x − 1)², so x + 3 = x² − 2x + 1.

Step 3: Rearrange into a quadratic equation. Subtract x + 3 from both sides: 0 = x² − 2x + 1 − x − 3, so 0 = x² − 3x − 2.

Step 4: Solve the quadratic. The expression x² − 3x − 2 does not factor neatly with whole numbers, so use the quadratic formula.

Using the quadratic formula x = [3 ± √(9 − 4(1)(−2))] / 2 = [3 ± √(9 + 8)] / 2 = [3 ± √17] / 2.

So the possible solutions are x = (3 + √17) / 2 and x = (3 − √17) / 2.

Step 5: Check for extraneous solutions by substituting back into the original equation √(x + 3) = x − 1.

First, approximate √17 ≈ 4.12. Then (3 + √17) / 2 ≈ (3 + 4.12) / 2 ≈ 7.12 / 2 ≈ 3.56. For x ≈ 3.56, the left side is √(3.56 + 3) ≈ √6.56 ≈ 2.56, and the right side is 3.56 − 1 = 2.56, so this value works.

Next, (3 − √17) / 2 ≈ (3 − 4.12) / 2 ≈ −1.12 / 2 ≈ −0.56. For x ≈ −0.56, the left side is √(−0.56 + 3) = √2.44, which is positive, about 1.56. The right side is −0.56 − 1 = −1.56, which is negative. The sides do not match, so x = (3 − √17) / 2 is an extraneous solution.

Answer: The solution is x = (3 + √17) / 2, and x = (3 − √17) / 2 is extraneous.

How radicals show up in later math

In middle and high school, radicals move from simple square roots to full radical expressions, such as 3√5 or ∛(x²). Students learn to simplify these expressions, add and subtract like radicals, and multiply radicals using rules about exponents and roots.

Radicals also appear in radical equations, where the variable is inside a radical sign. Solving these often involves isolating the radical, raising both sides to a power, and then checking for extraneous solutions that appear because of the squaring step.

Later, your child will study radical functions, such as f(x) = √x or f(x) = ∛(x − 4) + 2. They will analyze the domain (allowed x-values), range (possible outputs), and graphs of these functions, and compare them to other types like a linear function or an exponential function.

Common mistakes with radicals

Mixing up √(a + b) with √a + √b. For example, thinking √(9 + 16) = √9 + √16, so √25 = 3 + 4 = 7. The correct value is √25 = 5, and in general √(a + b) ≠ √a + √b.
Dropping the radical sign when simplifying. A student might write √50 = 25 instead of √50 = √(25 × 2) = 5√2. The radical sign must stay with any part that is not a perfect power.
Forgetting to check for extraneous solutions in radical equations. After squaring both sides, a student might accept all algebraic solutions, even ones that make the original radical expression undefined or produce a false statement. Every solution from the squared equation must be checked in the original equation.

Where radicals appear in everyday life

Radicals show up whenever a quantity depends on an area, volume, or power. For example, finding the side length of a square from its area uses a square root, and figuring out the length of a ladder leaning against a wall can use a square root from the Pythagorean theorem. Scientists and engineers also use radicals when working with formulas for speed, energy, and sound.

Related math terms to explore

Radicals connect to several other algebra ideas your child may study:

What is a radical?