Algebra/Roots and Radicals

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[edit] 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:

$\sqrt{x} = y \mbox{ if } y^2 = x \,$

For example, $\sqrt 9 = 3$ since $3^2 = 3\cdot3 = 9.$

[edit] 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:

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

[edit] 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 $b^{n} = a \,$ .

[edit] Irrational numbers

If you square root a whole number which is not itself the square of a rational 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 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 in decimal notation exactly. Writing the square root of 3 or any other non-square number as $\sqrt{3}$ is the simplest way to represent the exact value.