Algebra/Roots and Radicals

Square root

If you take the square root of a number the result is the number which when squared gives the first number. In algebra this can be written thus:

$\sqrt{x}=y\text{ if }{y}^2=x$

For example, $\sqrt{9}=3$ since $3^2=3\cdot 3=9.$

Cube roots

Roots do not have to be square. One can also take the cube root of a number ($\sqrt[3]{}$ ) of a number is the number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. For example, the cube root of 8 is 2, because $2\cdot 2\cdot 2=8$, or:

$\sqrt[3]{8}=2.$

Other roots

There are an infinite number of possible roots all in the form of $\sqrt[n]{a}$ which corresponds to $a^{\frac{1}{n}}$, when expressed using indices. If $\sqrt[n]{a}=b$ then $b^n=a$.

Irrational numbers

If you square root a whole number which is not itself a square number the answer will have an infinite number of decimal places. Such a number is described as irrational and is defined as a number which cannot be written as as a rational number: $\frac{a}{b}$, where a and b are integers.

However, using a calculator you can approximate the square root of a non-square number:

$\sqrt{3}\approx 1.73205080757$

The result of taking the square root is written with the approximately equal sign $\approx$ because the result is an irrational value which cannot be written exactly. Writing the square root of 3 or any other non-square number as $\sqrt{3}$ is the only way to represent the exact value.