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Square root - Meaning, Formula and Important Properties

 

Before learning about squares and square roots, which are special types of exponents, let us learn about the mathematical term ‘exponent’, which is the base of these concepts.

The exponent of a number shows how many times the number is multiplied by itself. For example, if we multiply 5 four times with itself, we can express it as 5 x 5 x 5 x 5. However, there is a much simpler and organized way to display this, i.e., 5 4 . Here ‘5’ is the base number and ‘4’ is the power or exponent of the base.

In the same way, if a number is multiplied twice with itself, it is known as the square of that number and the converse of it is known as the square root. Let us learn about them in detail: Square: Square is generally a number that is the product of an integer multiplied once with itself. For instance, if ‘k’ is the number, then k x k is known as the square of ‘k’. Square is usually represented by the number as the base raised to the power 2, i.e., k 2 .

Square root: The square root of a number is the counterpart of squaring a number. We know that the square of a number is the value of the number multiplied by itself to obtain the original number. Whereas the square root of a number is the number that is multiplied by itself to get the original number. If 'k' is the square root of 'l,' then k x k = l. The value of a square is always positive, but the square root of a number can yield two values, one positive and another negative. The negative numbers square to form positive numbers. For instance, the square of (-11) 2 = 121, which is a positive integer.

The square root is denoted by the ‘√’ symbol. This is well known as the radical symbol, and the term (integer) inside it is usually known as the radicand. The square root can be defined as the number raised to the power ‘1/2’ in terms of exponents. Let x be any positive integer; therefore, the square root will be (x 1/2 ). For example, (36) 1/2 or √36 is + 6 and – 6, respectively.

Below listed are some important properties of the square root:

  • Some numbers having an even number of zeros have square roots, while the numbers having an odd number of zeros do not have perfect square roots. For example, √100 is 10 while √1000 is 31.622.
  • The values inside two square root values can be multiplied. For instance, √x multiplied by √y equals √xy.
  • If two numbers having the value of square roots are multiplied with one another, the result is the radicand. For example, when √k is multiplied by √k, the result is k (without the radical sign).
  • A perfect square can never be negative (because the square of negative numbers is positive); thus, the square root of a negative number is imaginary and comes under complex mathematics.
  • The numbers ending with 2, 3, 7, or 8 in the unit's place is not the perfect square of any number. Therefore, they give decimal values of the square root while the numbers terminating with the numbers 1, 4, 5, 6, or 9 in the unit digit have a perfect square root. Thus, these numbers yield non-decimal perfect integer values.
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