Square Roots

Squaring, and undoing it

Squaring a number means multiplying it by itself. The results, starting from $1$, are the perfect squares:

A perfect square is any whole number you can reach by squaring a whole number, so $0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100$ are the first eleven of them ($0^2 = 0$ counts, even though it takes no work to see). The square root of a number asks the reverse question. Given a number, what do you square to get it? The square root of $36$ is $6$, because $6^2 = 36$. The square root of $100$ is $10$, because $10^2 = 100$.

We write a square root with the radical sign $\sqrt{\phantom{x}}$. The number tucked under it is called the radicand. So

read aloud as “the square root of twenty-five is five.” Squaring and square-rooting are inverse operations: on numbers that are zero or more, each one undoes the other. Square $5$ to get $25$, then take the square root of $25$ and you are back to $5$. Starting from a negative number the round trip does not bring it back. Squaring $-5$ gives $25$, and the square root of $25$ is $5$, not $-5$. In general the square root of $x^2$ is the size of $x$, which means $x$ with any minus sign dropped.

Because the radical undoes the square, the two cancel when you stack them. Start from $\sqrt{7^2}$: the radical hands back the number that was squared, so $\sqrt{7^2} = 7$. You never need the middle step of working out that $7^2$ is $49$ and then rooting $49$ instead. Stacking them in the other order lands in the same place, since $\left(\sqrt{49}\right)^2 = 7^2 = 49$. So for any non-negative number $a$,

Why the root is the non-negative one

There is a subtlety hiding in the question “what squares to $25$?” The obvious answer is $5$. But $-5$ also works, because a negative times a negative is positive:

So two different numbers, $5$ and $-5$, both square to $25$: the equation $x^2 = 25$ has both $x = 5$ and $x = -5$ as solutions. If the radical sign returned both, then $\sqrt{25}$ would not name a single number, and you could not safely write it inside a calculation. To keep the square root a single, well-defined value, mathematicians make a choice: the radical sign always returns the non-negative root. This chosen value is called the principal square root, so $\sqrt{25} = 5$ names one number even though the equation $x^2 = 25$ has two solutions. The same choice works for any positive number, and the one radicand that needs no choice at all is $0$ itself, since $0^2 = 0$ gives $\sqrt{0} = 0$.

Roots of perfect squares

When the radicand is a perfect square, the root is a whole number, and you find it by recognizing which number was squared. It pays to know the first several perfect squares by sight, because then their roots come for free:

Reading the table from the bottom row up gives you the square root directly. Since $144$ sits under $12$, you know $\sqrt{144} = 12$. Since $49$ sits under $7$, you know $\sqrt{49} = 7$.

In the figure below you set the two sides of a rectangle, and the readout counts the unit squares it holds.

Make the width and the height equal and you have built a square, so its area is one side multiplied by itself. Step that square up from $2$ by $2$ and you will generate the start of the table above ($4$, then $9$, $16$, $25$, $36$). In each of those squares, the side length is the root of the area. Then break the equality on purpose: set the rectangle $9$ wide and $3$ tall, and its area of $27$ lands between two perfect squares. Those squares are $25$ and $36$, and that trapping is exactly how this lesson later estimates a root that is not a whole number. It is only the square’s side, not just any rectangle’s, that is the square root of the area: an unequal rectangle can land on a perfect-square count too, like $2$ by $8$ giving $16$, without either side being $\sqrt{16}$.

The product of two perfect squares is itself a perfect square, and its root is the product of the roots. For instance $\sqrt{400} = 20$ because $400 = 4 \times 100$ and $\sqrt{4} \times \sqrt{100} = 2 \times 10 = 20$.

The product rule for square roots

Square roots of a product can be split apart. For non-negative numbers $a$ and $b$,

This is true because squaring the right-hand side returns $a \times b$. So the right-hand side is the non-negative number whose square is $a \times b$, which is exactly what $\sqrt{a \times b}$ means. Check it on $9$ and $49$: their roots are $3$ and $7$, and squaring that product puts both radicands back, $(3\times7)^2 = 3\times3\times7\times7 = 9\times49$, so $\sqrt{9\times49} = \sqrt{9}\times\sqrt{49}$.

Division splits the same way: $\sqrt{\tfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$, provided $a$ is zero or more and $b$ is greater than zero. The same squaring argument carries it, since $\left(\dfrac{\sqrt{a}}{\sqrt{b}}\right)^2 = \dfrac{a}{b}$. So a quotient of square roots combines or splits just like a product.

This rule lets you simplify a root by pulling out any perfect-square factor hiding inside the radicand. The trick is to split the radicand into a perfect square times whatever is left.

Estimating roots that are not whole numbers

Most whole-number radicands are not perfect squares, so their roots are not whole numbers. You can still pin a root down closely by trapping it between the two perfect squares it falls between. Take $\sqrt{10}$. The perfect squares on either side of $10$ are $9$ and $16$:

Taking the square root of all three keeps the order, because a bigger number has a bigger root:

You can say more: since $10$ is much closer to $9$ than to $16$, its root sits close to $3$, just a little above it. A quick check confirms that estimate: $3.2^2 = 10.24$ is already a bit over $10$, so $\sqrt{10}$ is a little under $3.2$. (A calculator gives $\sqrt{10} \approx 3.16$.)

The method is general: to estimate the square root of any whole number that is not a perfect square, find the nearest perfect square below the radicand and the nearest one above it. The root lies between their two whole-number roots, and it leans toward whichever perfect square the radicand is closer to.

Why most square roots are irrational

When you estimate $\sqrt{10}$ you can keep refining: $3.1$, then $3.16$, then $3.162$, and the decimal never settles and never falls into a repeating pattern. A square root of a whole number that is not a perfect square is an irrational number. Its decimal expansion runs on forever without ever repeating, so it can never be written exactly as a fraction or as a terminating decimal.

You met this idea when fractions and decimals were compared. A fraction always turns into a decimal that either stops or repeats, while an irrational number does neither. Numbers like $\sqrt{2} = 1.41421356\ldots$ and $\sqrt{3} = 1.7320508\ldots$ are the classic examples. The impossibility can be proved, as the Greeks first did for $\sqrt{2}$ (the history below tells that story). This is why we usually leave such a root in its exact radical form, writing $\sqrt{2}$ rather than a rounded decimal.

Common mistakes