The following important points should be kept in mind while we work with perfect squares. The final answer for the square of 65 is 4225. Further, for the final step write the digits of the second step, followed by 25. Now for the third step, square the number 5 to get 25. First, we need to separate the numbers 6 and 5. Now, we can find the square of 65 through a sequence of four simple steps. Numbers Ending with Digit 5: Let's consider a number ending with 5, like 65. In other words, this can be used to calculate the square of a large number without using the long multiplication method. These can be applied to square a number in a very short time. You can find the square of a number by multiplying it by itself, for example, 6 × 6 = 36, However, there are some simple methods that work for special types of numbers. We can see that 9 is a whole number, therefore, 81 is a perfect square. Let us take another example of the number 81. As we can see, 4.89 is not a whole number, so, 24 is not a perfect square. For example, to check whether 24 is a perfect square or not, let us calculate its square root. If the square root is not a whole number, then the given number is not a perfect square. If the square root is a whole number, then it is a perfect square. Another Way to Identify Perfect SquaresĪnother way to check whether a number is a perfect square or not is by calculating the square root of the given number. 400 and 300 both have an even number of zeros at the end, but 400 = 20 2, which is a perfect square, but 300 is not a square of any whole number. If there are an even number of zeros, then it might be a perfect square. If there is an odd number of zeros, then it's definitely not a perfect square. If the number ends with the digit 0, then you may look for the following: How many zeros are there at the end of the number? Let's say we have a number 1000. The numbers 159 and 169 both end with the digit 9 but 169 is a perfect square, whereas 159 is not. Let us look at a few deviations from the above-defined rules of a perfect square number. The numbers ending with 1 and 9 will have 1 as the units place digit in its square number.The number ending with 2 and 8 will have 4 as the units place digit in its square number.The number ending with 4 and 6 will have 6 as the units place digit in its square number.The number ending with 5 will have 5 as its units place digit in its square number.The numbers ending with 3 and 7 will have 9 as the units place digit in its square number.The following observations can be made to identify a perfect square. Numbers that have any of the digits 2, 3, 7, or 8 in their units place are non-perfect square numbers, whereas, numbers that have any of the digits 0, 1, 4, 5, 6, or 9 in their units place might be perfect squares. After trying various perfect square numbers you would have observed an important property of perfect squares. You will notice that they end with any one of these digits 0, 1, 4, 5, 6, or 9. Observe the last digit of the perfect square numbers 1 to 20 as given in the table above. Other examples of a perfect square trinomial are y 2 -8y+16 and 4x 2+ 12x +9. Here, y 2 +6y+9 is a perfect square trinomial. For example, if we square the expression (y+3), we use the identity, (a+b) 2= a 2 +2ab+b 2, and we get, (y+3) 2 = y 2 +6y+9. Perfect Square Trinomial: An expression that is obtained from the square of a binomial is termed a perfect square trinomial. Now, that we know about perfect squares, let's learn about perfect square trinomials. We can see that 8 is a whole number, therefore, 64 is a perfect square. Let us take another example of the number 64 → √64 = 8. As we can see, 4.58 is not a whole number/integer, so, 21 is not a perfect square number. For example, to check whether 21 is a perfect square or not, let us calculate its square root. If the square root is a whole number, then the given number is a perfect square, but if the square root value is not a whole number, then the given number is not a perfect square number. To know whether a number is a perfect square or not, we calculate the square root of the given number. This can be understood in another way with the help of square roots. Here, 81 is a perfect square because it is the square of a whole number, 9. Let us substitute the formula with values. So, the perfect square formula can be expressed as: Let us assume if N is a perfect square of a whole number x, this can be written as N = the product of x and x = x 2.
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