Radical Expressions

Radical expressions are algebraic expressions that involve roots. The most common is the square root, but there are also cube roots and higher-order roots.

The root operation gives the number that, when multiplied by itself, returns the original number.

Properties

Only non-negative numbers have real square roots. For example, the square root of -4 has no real result.

If two numbers are non-negative, the square root of their product can be broken down into the roots of the factors.

For a non-negative numerator and positive denominator, the square root of a fraction can be written as the separate roots of the numerator and denominator.

The nth root of a number raised to the nth power returns the original number.

Simplification

If a perfect square or higher power is under the root, the expression can be simplified by taking the perfect square or higher power out in front of the root.

Here, 25 is a perfect square, so it can be taken out in front of the root, and the remainder stays inside.

For higher-order roots, we proceed similarly: 27 is a perfect cube (3³), so it can be taken out in front of the cube root.

Rationalizing the Denominator

In mathematics, it is often required that the denominator has no root. In such cases, we rationalize: multiply the fraction so that the root disappears from the denominator.

This procedure ensures that the denominator remains in a simple, root-free form.

Higher-Order Roots

Root extraction is not limited to square roots. The cube root, for example, is the inverse of the third power.

In general: the nth root of a number is the number that, when raised to the nth power, returns the original value.

Practical Examples

Radical expressions appear in many fields outside of mathematics as well.

• In geometry: the diagonal of a square is the side length multiplied by the square root of 2.
• In physics: according to the Pythagorean theorem, the hypotenuse of a right triangle is the square root of the sum of the squares of the legs: \(\sqrt{a^2 + b^2}\).
• In statistics: the square root always appears in variance calculations.

Practice Exercise