Squares and Square Roots

Finding Squares of Given Numbers

Suppose you are asked to multiply a number with itself. What are you doing? What is the product known as? You are squaring a number. The multiplication of a number by itself is the squaring of a number. The product is the square of the number. Here you will learn about finding square numbers. You can find the squares of all number. Let’s start to learn about finding squares of given numbers.

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Square Numbers

Suppose you have a square of length l. What is the area of that square? The area is calculated as l × l = 2. This 2 is the square of the length of the side of the square. Just like the length of the sides of a square are all equal. For finding the squares of a number we multiply the number by itself only. A square number is always positive.

The numbers like 4, 9, 25 and others can be expressed as the product of a number by itself. 2 × 2 = 2 expresses the number 4, 3 × 3 = 32 expresses 9 etc. The numbers 1, 4, 9, 25, 36, 49 etc. are the square numbers. These are sometimes known as perfect squares.

Square Numbers

Suppose we have a number 72. Is 72 a square number? We know that 82 = 64 and 9= 81. If 72 is a square number, it must be the square of a number between 8 and 9. But there is no natural number between 8 and 9. So, 72 is not a square number. Moreover, 72 = 8 × 9 and not 8 × 8 or 9 × 9. So, 72 is not a square number.

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Square Numbers

Properties of Square Numbers

Suppose your friend asked you to guess any square number and he will tell the unit place digit of the number. It is interesting to know that all the square numbers end with 0, 1, 4, 5, 6 or 9. None of the perfect squares end with 2, 3, 7 or 8 at unit’s place. Let us know more of these properties

  • If a number has 0 in the unit’s place, then its square ends in 0.
  • If a number has 1 or 9 in the unit’s place, then its square ends in 1.
  • The square, if, ends in 4, then the number has 2 or 8 in the unit’s place.
  • The number will have 3 or 7 in the unit’s place if the square ends in 9.
  • The number will have 4 or 6 in the unit’s place if the square ends in 6.
  • The square will end in 5 if the number has 5 in the unit’s place.
  • A number ending with one zero will results in 2 zeros in its square. Those ending with two zeros will square itself and give four zeros and so. This means that a perfect square will always end in even number of zeros.

Solved Examples for You

Question 1: What will be the unit’s digit in the square of the following numbers?

  • 12487
  • 1324
  • 91478
  • 1251

Answer : The unit’s digit in the square of the following is:

  • 12487 is 9 (as 72 = 49. 9 in the unit’s place).
  • 1324 is 6 (as 42 = 16. 6 in the unit’s place).
  • 91478 is 4 (as 82 = 64. 4 in the unit’s place).
  • 1251 is 1 (as 12 = 1. 1 in the unit’s place).

Question 2: Comment on the square of an even number and of an odd number?

Answer : The square of an even number is always an even number and the square of an odd number is always an odd number. The square of an even number will always have 4, 6, or even number of zeros in its unit’s place. And the square of an odd number will always have 1, 5 or 9 in its unit’s place.

Question 3: What are the different square numbers starting from 1 to 100?

Answer: When we multiply a whole number with equal times itself, then the resulting product is known as a square number or a perfect square. So, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are all the square numbers between 1 and 100.

Question 4: Is 25 a perfect square?

Answer: As we know that ‘25’ is a natural number and the square root of the number ‘25’ is a natural number ‘5’. Therefore, yes, ‘25’ is a perfect square.

Question 5: What is the perfect square formula?

Answer: When a polynomial is multiplied with itself, then the result is said to be a perfect square. For instance, this polynomial ‘ax2 + bx + c’, if b2 = 4ac is a perfect square.

Question 6: What is a perfect square trinomial?

Answer: A trinomial is said to be a perfect square trinomial only if it is factorable into a binomial multiplied with itself. In a perfect square trinomial, 2 of our terms will be the perfect squares. Such as, in the trinomial ‘x2 – 12x + 36’, here, both x2 and 36 are said to be the perfect squares.

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