Roots

A root is the inverse of raising to a power. The square root of a number $a \ge 0$ is the non-negative number $b$ with $b^2 = a$:

For example $\sqrt{49} = 7$, because $7^2 = 49$.

The condition $a \ge 0$ is not incidental: the square of every real number is non-negative, so a negative number has no square root among the real numbers.

Cube roots behave differently:

Here a negative input is fine, because the cube of a negative number is negative: $\sqrt[3]{-8} = -2$.

Arithmetic with roots

Roots distribute over multiplication and division:

But not over addition: $\sqrt{9 + 16} = \sqrt{25} = 5$, while $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$.

Taking a factor out of a radical

Most roots are not rational numbers — $\sqrt{72}$ cannot be written as a fraction. It can, however, be simplified: spot a square factor under the radical and take it out:

Absorbing a factor under a radical

The same law read from right to left tucks a factor under the radical. It goes in squared:

For example $3\sqrt{5} = \sqrt{3^2 \cdot 5} = \sqrt{45}$. The condition $a \ge 0$ is necessary: what sits under a square root cannot be negative, so a minus sign stays outside. The expression $-2\sqrt{3}$ is $-\sqrt{12}$, not $\sqrt{12}$.

Taking out and absorbing are two directions of one move. Taking out simplifies ($\sqrt{72} = 6\sqrt{2}$); absorbing brings expressions to a common form — which is exactly what comparing or adding two roots needs.

Estimating a root

Most roots have no exact decimal form, but every one can be trapped between two whole numbers in seconds. Find the two consecutive squares the number under the radical lies between:

For $\sqrt{30}$ those squares are $25 = 5^2$ and $36 = 6^2$, so $5 < \sqrt{30} < 6$. The greatest integer not exceeding $\sqrt{30}$ is therefore $5$.

The estimate can be sharpened without a calculator: $30$ is not far from the middle of the stretch from $25$ to $36$, so $\sqrt{30} \approx 5.5$. Checking by multiplication, $5.5^2 = 30.25$ — slightly too much, so the true value is a little smaller ($5.477\ldots$).

Estimating is above all a check on an answer. If a calculation or a calculator produces $\sqrt{30} = 15$, it is enough to notice that $15^2 = 225$, not $30$.

Comparing roots

Taking a root preserves the order of numbers: the larger the number under the radical, the larger the root.

So $\sqrt{17} > \sqrt{15}$, and no arithmetic is needed. The difficulty starts when the roots also carry factors in front — then you absorb them first, so that one number is compared with one number:

The gap here is $\sqrt{45} - \sqrt{44} \approx 0.075$ — by eye the two are indistinguishable, and the rule settles it with certainty. Comparing by the factor in front alone, or by the number under the radical alone, fails: $2\sqrt{11}$ has the smaller factor and the larger number underneath, and is smaller all the same.

Rationalising a denominator

A fraction with a root in the denominator is usually rewritten by multiplying the numerator and the denominator by that same root — that is, by one:

Hence the familiar $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ — the same number, a handier form.

A root is a power

A root is simply a power with a fractional exponent:

That ties the two topics together: every law of exponents then holds for roots as well, with no separate rules to memorise.

Common mistakes

• Splitting a root over addition — $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$.
• Writing $\sqrt{16}$ as $\pm 4$ — a square root is non-negative by definition; the $\pm$ appears only when solving the equation $x^2 = 16$.
• Stopping at the approximation — when a question asks for the exact value, the answer is $6\sqrt{2}$, not $8.49$.
• Looking for the square root of a negative number — it does not exist among the reals, unlike the cube root.
• Absorbing a factor without squaring it — $3\sqrt{5}$ is $\sqrt{45}$, not $\sqrt{15}$.
• Pulling a minus sign under the radical — $-2\sqrt{3}$ is $-\sqrt{12}$; a square root cannot have a negative number under it.
• Comparing roots by the factor in front — $3\sqrt{5} > 2\sqrt{11}$ even though $5 < 11$; only the absorbed forms $\sqrt{45}$ and $\sqrt{44}$ settle it.
• Estimating "halfway between the squares" — $\sqrt{x}$ grows more and more slowly, so for an $x$ halfway from $n^2$ to $(n+1)^2$ the root lands slightly above the midpoint between $n$ and $n+1$.