LESSON 10
SQUARE ROOTS
The Inverse of Squaring a Number
What is a Square Root?
A square root is the inverse operation of squaring a number. If 5² = 25, then the square root of 25 is 5. We write this as √25 = 5. The symbol √ is called a radical sign.
Square roots ask the question: what number multiplied by itself gives this result? They appear in geometry, physics, and algebra, and understanding them is essential for working with the Pythagorean theorem and quadratic equations.
Perfect Squares
A perfect square is a number whose square root is a whole number. The first several perfect squares are easy to memorize and recognize. Knowing them makes mental math much faster and helps you estimate square roots of other numbers.
Estimating Non-Perfect Square Roots
Most numbers are not perfect squares, so their square roots are decimals that go on forever. To estimate, find the two perfect squares the number falls between. The square root will be between those two whole numbers, and you can narrow it down further by testing decimals.
Simplifying Square Roots
To simplify a square root, find the largest perfect square that divides evenly into the number under the radical. Write it as a product, take the square root of the perfect square factor, and leave the rest under the radical sign.
The Pythagorean Theorem
One of the most important uses of square roots is the Pythagorean theorem: in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (a² + b² = c²). To find a side length, we take a square root.
Key Facts
Every positive number has two square roots — a positive and a negative one. √25 = 5 and −5, since both (5)² and (−5)² equal 25.
The square root of a negative number is not a real number. It belongs to a system called imaginary numbers, written with the symbol i.
The Pythagorean theorem (a² + b² = c²) was known to Babylonian mathematicians over 1,000 years before Pythagoras was born.
√2 is irrational — its decimal expansion never ends or repeats. It was one of the first numbers proven to be irrational, by ancient Greek mathematicians.