What is Square Root?

Many mathematical operations have an inverse, or opposite, operation. Subtraction is the reverse of addition, the division is the inverse of multiplication, and so on. Squaring, which we learned more about in a previous lesson, has an inverse too, called Square Root.

What is Square Root?

The square root of a number n, written as \dpi{120} \fn_jvn \large \sqrt{n}  is the value that gives n when multiplied by itself.

42 = 4⋅4 = 16

We know that 16 is the square of 4. The square of -4 is 16 as well

(−4)2=(−4)⋅(−4) = 16

4 and -4 are known to be the square roots of 16.

All positive real numbers have two square roots, one positive square root, and one negative square root. The positive square root is sometimes described as the principal square root.

The main reason that we have two square roots is shown above. The product of two numbers is positive if both numbers have the same sign as holds true with squares and square roots

n2 = nn = (n)(n)

How to write the square root of a number?

A square root is written with a radical symbol \dpi{120} \fn_jvn \large \sqrt{n}

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The value or expression under the radical below denoted “n”, is known as the radicand.

\dpi{120} \fn_jvn \large \sqrt{n}

To show that we need both the positive and the negative square roots of a radicand we put the symbol ± (called as plus-minus) before the root.

±\dpi{120} \fn_jvn \large \sqrt{16}± 4


Zero has only one square root which is 0 itself.

\dpi{120} \fn_jvn \large \sqrt{0}= 0

Non-positive numbers don’t have real square roots because a square is either positive or 0.

Perfect square

If the square root of a positive integer is another integer then the square is called a perfect square.

For example

36 is a perfect square as

\dpi{120} \fn_jvn \large \sqrt{36}± = ±6


Approximation method

If the number is not a perfect square i.e. the square root is not a whole number then you have to find the square root by approximation method.

\dpi{120} \fn_jvn \large \sqrt{5}± = ±2.2360\dpi{120} \fn_jvn \large \approx 2.23

The square roots of numbers that are not a perfect square are members of the irrational numbers. This means that they can’t be written as the quotient of 2 integers. The decimal type of an irrational number will neither end nor repeat. The irrational numbers together with the rational numbers constitute the real numbers.

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