A hardware store sells [latex]16[/latex]-ft ladders and [latex]24[/latex]-ft ladders. A window is located [latex]12[/latex] feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground [latex]5[/latex] feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown, and use the Pythagorean Theorem.

[latex]\begin{align} {a}^{2}+{b}^{2}& = {c}^{2} \\ {5}^{2}+{12}^{2}& = {c}^{2} \\ 169& = {c}^{2} \end{align}[/latex]

Now we need to find out the length that, when squared, is [latex]169,[/latex] to determine which ladder to choose. In other word we need to find a square root. In this section we will investigate methods of finding solutions to problems such as this one.

Square Roots

When the square root of a number is squared, the result is the original number. Since [latex]{4}^{2}=16[/latex], the square root of [latex]16[/latex] is [latex]4[/latex]. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.

When we talk about square roots in math, we usually mean both the positive and negative numbers that, when multiplied by themselves, give us the original number. In math classes, especially when you’re solving problems or learning algebra, it’s important to think about both these answers unless we’re told to find just one. This helps us understand all the possible solutions to an equation, which is a big part of learning how to solve math problems correctly.

As we explore square roots further, there’s a special version we often use called the principal square root. This refers specifically to the positive square root of a number. It’s helpful to know about this because it’s commonly used in many situations.