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Square Roots and Radicals

A square root is defined as a number which when multiplied by itself gives a real non-negative number called a square.

A square root is best defined using geometry where, considering a square (which is a four sided polygon whose sides are all equal), a square root is defined as the length of the diagonal of this square (a diagonal is a line drawn from one vertex/corner to the opposite vertex of the square).

A radical is a root of a number. A square root is a radical. Roots can be square roots, cube roots, fourth roots and so on.

A square root is commonly shown as

where is known as the radical sign and is known as the radicand.

A square root of a number can also be represented as

and a radical as

where we say that in the above, we're finding the nth root of x. For more on the above notation, refer to section on exponents.

A radical can also be represented as

A square root is also represented as

A cube root as

A fourth root as

Every square has two square roots; one positive and the other negative. This is shown as:

which is written as

This can be proved in the following way. Consider a number, a

but also

the latter is because a negative multiplied by a negative equals a positive.

And so it follows that

For example,

but also

Therefore,

Thus it follows that any real positive number has two roots. But when talking about radicals

in other words, only refers to +x which is known as the principal square root. So despite having said above that

we usually only consider

especially if the is used.

But if the question asked is in the form

always give both the positive and negative roots, i.e.

Although any real positive number can be considered a square number and thus has a square root, we only consider numbers with whole number square roots as squares.

For example

Properties of Square Roots and Radicals

Properties of square roots and radicals guide us on how to deal with roots when they appear in algebra.

Examples of Square Roots and Radicals

Evaluate the following:

1.

Solution:

2.

Solution:

3.

Solution:

4.

Solution:

The above is left as is, unless you are specifically asked to approximate, then you use a calculator.

5.

Solution:

Quiz on Square Roots and Radicals

1. Simplify the following expression without using a calculator

A.
B.
C.
D.
The correct answer here would be C.

The answer is obtained as follows:

First factor out both the numerator and denominator into numbers whose square roots are easy to find

From here you can cancel the terms that appear in both the numerator and denominator:

See Radical Functions in Pre-Calculus for help with functions involving square roots and radicals.

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