Welcome to learning about square roots! A square root is the inverse operation of squaring a number. If y squared equals x, then y is the square root of x. The symbol with the radical sign represents the positive square root. For example, since 4 squared equals 16, the square root of 16 is 4.
Perfect squares are special numbers whose square roots are whole numbers. Let's look at the first five perfect squares. One squared equals one, so the square root of one is one. Two squared equals four, so the square root of four is two. Three squared equals nine, so the square root of nine is three. Four squared equals sixteen, so the square root of sixteen is four. Five squared equals twenty-five, so the square root of twenty-five is five.
When dealing with non-perfect squares, we can often simplify them by factoring out perfect squares. The method is to factor the number under the radical, find any perfect square factors, and take the square root of those perfect squares outside the radical. For example, the square root of twelve equals the square root of four times three, which equals the square root of four times the square root of three, which simplifies to two times the square root of three. Similarly, the square root of eighteen equals three times the square root of two, and the square root of twenty equals two times the square root of five.
Now let's learn how to solve equations that contain square roots. The process involves four key steps. First, isolate the square root term on one side of the equation. Second, square both sides to eliminate the square root. Third, solve the resulting equation for the variable. Fourth, and this is crucial, always check your answer in the original equation. Let's work through an example. We want to solve the square root of x plus three equals five. Step one: the square root is already isolated. Step two: we square both sides to get x plus three equals twenty-five. Step three: solving for x gives us x equals twenty-two. Step four: we check by substituting back into the original equation. The square root of twenty-two plus three equals the square root of twenty-five, which equals five. Our answer checks out!
To summarize what we've learned about square roots: Square roots are the inverse operation of squaring numbers. Perfect squares like four, nine, and sixteen have whole number square roots. Non-perfect squares can often be simplified by factoring out perfect square factors. When solving equations with square roots, always remember to check your solutions in the original equation. Square roots are fundamental concepts that appear throughout mathematics and have many practical applications.