Hey everyone, welcome back to App Like. Although we post only technology updates today, we are doing something out of the box. It will help programmers, coders to troubleshoot the mathematical algorithms. And coders can check their math skills too. Today, we will explore a perfect square and the relationship between square roots and a perfect square.
So we get to infographic explore that question, what exactly is a square root? Let’s go ahead and take a look.
A perfect square is a number that can be expressed as the product of two integers or as the second exponent of an integer. 25 is a perfect square, for example, because it is the product of the integer five multiplied by itself, 5* 5 equals 25. However, 21 is not a perfect square number because it cannot be expressed as the product of two equal integers.
So we will start our discussion by looking at a rectangle (Look at the above picture). And this rectangle has a width of four units and a length of eight units. Now we’re going to envision this in terms of area.
So that width of four, think of that as four equal-sized units that will split up along the width. And then, for the length, we have eight equal-sized units, which we’ll split up along the length. And now we have these squares that make up the inside the area of that figure, length times width, in this case, four times eight, which we know is 32. So the area of this rectangle is 32 square units. (See below picture)
So now, let’s look at what would happen if we cut this rectangle in half. So now, instead of it being a four by eight
rectangle, it’s a four by four, which makes it a square; since all the sides have the same length, the area of this square is four times four, I know is equal to 16 square units.
And this relationship allows us to say that 16 is a perfect square. And also that the square root of 16 is four, since four squared equals 16. And we can think of this relationship every time we think about perfect squares. For example, if we took this four by four square and cut issued in half to make it eight by eight.
16 is a perfect square
Now, if we counted up all those squares to find the area or just multiplied eight by eight, eight squared, we would get a result of 64 square units. So 64 is also a perfect square. And the square root of 64 is eight because eight squared will equal 64.
So now we can say that a perfect square is a number that can be made by multiplying a whole number by itself by squaring any whole number. Cool.
How Do You Spot Perfect Squares?
So now we can revisit the two perfect squares from earlier. And that was the four by four square and the eight by eight square. Now four times four, if we think of this in terms of area, is 16, equal to four squared.
And eight times eight is equal to 64, which we know is equal to eight squared. Notice again that four and eight are both whole numbers.
If we take the square root of 16, our result is four. And if we take the square root of 64, our result is eight.
So the most important thing to take away from this lesson is the relationship between a perfect square and its square root.
Take note of the last digit of the perfect square numbers 1 to 20 in the table above. You’ll notice that they all end with one of the following digits: 0, 1, 4, 5, 6, or 9. You would have noticed an important property of perfect squares after experimenting with various perfect square numbers. Non-perfect square numbers have any of the digits 2, 3, 7, or 8 in their unit place, whereas perfect square numbers have 0, 1, 4, 5, 6, or 9 in their units place. To identify a perfect square, make the following observations.
1. The units placed digit in the square number ending in 3 and 7 will be 9.
2. The square number of a number that ends in 5 will have five as its units place digit.
3. The units place digit in the square number of a number ending in 4 and 6 is 6.
4. The units placed digit in the square number of a number ending in 2 or 8 will be 4.
5. The units placed digit in the square number of numbers ending in 1 and 9 will be 1.
Relationship between a perfect square and its square root
So the most important thing to take away from this lesson is the relationship between a perfect square and its square root. And now we can explore that relationship a little bit more; let’s take a look at an example of a perfect square of 25. And a non-perfect square 3025 is a perfect square. Because 25 is equal to five squared. And the square root of 25 is equal to five, which is a whole number.
So by definition, everything works out as a non-perfect square 30. If I take the square root of 30, I get approximately 5.48. Which, of course, is not a whole number, so there is no whole number that we can square to get to 30 the way that we can square five to get to 25. Now 25 and 30 are pretty close to each other in value. And if we think about these two numbers in terms of area. Again, we know that 25 is a perfect square with dimensions five by five.
And if we think about that area, compared to that 30, If we slide it over, we see that it kind of fits, but there is some room leftover that orange space, which doesn’t quite hold one full square. It only holds about half of a square, which is where that point four eight that decimal comes from. And again, this is because there is no whole number that we can square to get to 30. We’d have to have a decimal value to approximate a square root for a number like 30.
Perfect Square Hints and Techniques:
While working with perfect squares, keep the following important points in mind.
- A perfect square that ends with 0 has an even number of zeros at the end.
- Because (-ve) (-ve) = (+ve), perfect squares are always positive.
- Perfect squares’ square roots can be positive or negative.
- We can also find perfect cubes by multiplying a number three times with itself.
- We can calculate the square root of a given number to see if it is a perfect square or not. It is a perfect square if the square root is a whole number. The given number is not a perfect square if the square root is not a whole number.
What are Square Roots?
The square root of a number is defined as the inverse of squaring a number. The square of a number is the value multiplied by itself to get the original number. In contrast, the square root of a number is the number multiplied by itself to get the original number. If ‘a’ is the square root of ‘b,’ then a*a equals b. Because the square of any number is always a positive number, each has two square roots, one positive and one negative. For example, the numbers 2 and -2 are both square roots of 4. However, only the positive value is written as the square root in most cases. Are you curious to explore more about perfect squares and square roots, you may join Cuemath – Math’s best live tutoring platform.
Methods for Finding the Square Root of a Number
It is very simple to calculate the square root of a perfect square number. Perfect squares are positive numbers that can be written as a number multiplied by itself. In other words, perfect squares are the values of power 2 of any integer. To find the square root of a number, we can use one of four methods, which are as follows:
- Square Root by Repeated Subtraction Method
- Square Root by Prime Factorization Method
- Estimation Method Square Root
- Long Division Square Root Method
Remember that the first three methods are best suited for perfect squares, whereas the fourth method, long division, can be applied to any number, perfect or not.
Let see the few more perfect square expression
So now we’re going to go ahead and visually explore some of the most common perfect squares and their relationship to their square root.
- So one squared is one times one was just equals one.
- Two squared is two times two Which equals four,
- three squared is three times three, which equals nine,
- four squared, again, four times four, which equals 16.
- And again, these are all perfect squares, five squared is 25,
- six squared, six times six equals 36.
- Seven squared is seven times seven, which equals 49.
- Eight squared is eight times eight, which equals 64. We actually looked at that one earlier.
- And nine squared, nine times nine is equal to 81. And the square root for each one of these perfect squares is the value that you had the square to get it in the first place.
So, for example, the square root of four is equal to two, the square root of nine is equal to three, the square root of 16 is equal to four, and so on. And this is why raising a number to the second power, or the power of two, is referred to as squaring the number. So those are the basic concepts involved with perfect squares and square roots. So keep that in mind as you continue to build upon that understanding and apply it to algebra and more advanced levels of mathematics. And we’ll catch you guys next time, all right, so that’s it for that lesson. I hope you found it helpful.