What are Square Roots and Cube Roots?

Square Roots

Suppose we have an expression given by z2 = y. We can say the square root of a number y is a number such that the above equation is satisfied. Thus, when a number z is multiplied by itself, the product is a square number y. Square roots are both negative and positive. For example, the square root of 25 is 5 and -5. All non-negative real numbers m have a unique non-negative square root. This is referred to as the principal square root and is denoted by b, and is the radical sign. For example, 5 is the principal square root of 25 as 5 * 5 = 52 = 25 and 5 is a non-negative integer. 25 is referred to as the radicand. Thus, the radicand is the number whose square root is being considered.

History of Square Roots

The Yale Babylonian Collection YBC 7289 clay tablet created between 1800 BC to 1600 BC contains the square root of two representations. The Rhind Mathematical Papyrus in 1650 BC shows that Egyptian used an inverse proportion method to extract square roots. The Sulba Sutras in Ancient India, which was dated even before 800 – 500 BC, details several theoretical and applied aspects of squares and square roots. Aryabhata in Aryabhatiya gives a method for finding the square root of numbers consisting of several digits. Several other cultures and mathematicians, such as the Greeks and Chinese, came up with their own methods to find square roots. Eventually, all these excerpts were combined to come up with what we know as the modern version of finding square roots.

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Properties of Square Roots

  • A perfect square number will have a perfect square root
  • An even perfect square will always have an even square root
  • An odd perfect square will always have an odd square root
  • A perfect square cannot be negative; therefore, the square root of a negative number is not defined, or it comes under complex numbers
  • Numbers that end with 1, 4, 5, 6, and 9 will always have a square root
  • Numbers that end with 2, 3, 7, and 8 will not have a perfect square root
  • A square root is possible only for numbers that end with an even number of zeros. Thus, numbers having an odd number of zeros at the end will not have a square root
  • We can multiply two square roots
  • If two same square roots are multiplied, they give a non-square root number

Cube Roots

Cube roots of a number are similar to square roots. When a number n is multiplied by itself three times to give the number itself, it is called a cube root. It can be represented as n3 = m. Thus we can say the cube roots of a number m are those numbers n that satisfy the given equation. Cube roots can be negative and positive numbers. The cube roots of -125 are -5, -5, -5, as when we cube -5, we get -125.

Properties of Cube Roots

  1. The cube roots of odd numbers will always be an odd number
  2. The cube roots of even numbers will always be an even number
  3. The cube roots of a negative integer will always be negative

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