Welcome to our lesson on square roots. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. We can visualize this with a square that has sides of length 3 units, giving us an area of 9 square units.
Perfect squares are numbers that have whole number square roots. These are the results when we multiply a whole number by itself. The first few perfect squares are 1, 4, 9, 16, 25, and so on. We can visualize these as actual squares where each side length corresponds to the square root, and the total area gives us the perfect square number.
The square root symbol is called a radical, and it looks like this. The number under the radical symbol is called the radicand. When we write the square root of 16 equals 4, we're saying that 4 is the number that, when squared, gives us 16. Let's look at some more examples of square root notation.
Not all numbers are perfect squares. When we need to find the square root of a number like 8, we can estimate by finding which perfect squares it falls between. Since 8 is between 4 and 9, and we know that the square root of 4 is 2 and the square root of 9 is 3, we can conclude that the square root of 8 must be between 2 and 3. Using a calculator, we find that the square root of 8 is approximately 2.83.
Square roots have many practical applications, especially in geometry. One of the most famous uses is in the Pythagorean theorem, which helps us find the length of the hypotenuse in a right triangle. For example, if we have a triangle with sides of length 3 and 4, we can use the theorem: a squared plus b squared equals c squared. Substituting our values: 3 squared plus 4 squared equals c squared, which gives us 9 plus 16 equals c squared, so 25 equals c squared. Taking the square root of both sides, we get c equals the square root of 25, which equals 5.