The root is the inverse operation to the exponentiation. In other words, the square root is a factor of a number which will become the original number when multiplied by itself. For example, the square root of 4 is 2 but if 2 is multiplied by itself, it will result in the original number (which is 4). Another example is 49, the square root of 49 is 7 but if 7 is multiplied by itself then it will result in the original number (which is 49). Given two numbers, called the radicand and index, find a third number, which is called the root, such that when elevated to the index, it is equal to the radicand.

\sqrt [ index ]{ Radicand } =Root

The index for the square root is 2, although in this case it is omitted. Let's come back to all previous examples and describe them. In the first example, the number is 4 and the index will be 2, radicant will be 4 and the root will be 2. In the case of the second example, the number is 49 and the index is the same (which is 2). The radicant will be 49 and the root will be 7.

\sqrt { Radicand } =Root

The index usually defines how many times the number should be multiplied to get the original number. In the above examples, the index was taken 2 that means the number should be multiplied twice by itself to get the original number. Here is a new example with a different index, let's say you are asked to find \sqrt [ 3 ]{ 8 } which means you need to find three factors. If we multiply 2 three times, it will end up 8 hence \sqrt [ 3 ]{ 8 } is 2.

In simple words, the square root of a number, a, is accurate when a number is found, b, which when elevated to the square is equal to the radicand: { b }^{ 2 } = a

Exact square root

An exact square root has the remainder 0.

Radicand = { (Exact \quad root) }^{ 2 }

\sqrt { 16 } = 4 \qquad 16 = { 4 }^{ 2 }

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Perfect Square

These are the numbers that have exact square roots.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, ...

Not exact square root

If a number is not a perfect square, the square root is no exact.

Radicand = { (Root) }^{ 2 } + Remainder

\sqrt { 17 } = 4.123 \qquad 17 = { 4 }^{ 2 } + 1

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.