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The square root of a number is a value that, when multiplied by itself, gives the original number. The square root is essential in various fields such as construction, engineering, design, finance, and navigation. In this topic, we are learning about the square root.
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In mathematics, a square root is a fundamental concept. A number's square root is a value that yields the original number when multiplied by itself. Look at this formula (y2= x), now we can say that the square root of x is a number we get by multiplying y by itself (y2= x). Just the same, we can figure out the square root of any number. Any positive real number has one positive and one negative square root. For instance, the square root of 9 is ±3, where 3 x 3 = 9 also (-3) x (-3) = 9.
The history of square roots dates back thousands of years. It shapes young minds in understanding of the concept as we know it today. Ancient civilizations, like the Babylonians and Egyptians, were pioneers in developing the square roots of mathematics. They even developed efficient methods to approximate the value.
By around 500 BCE, the Pythagoreans employed square roots for their works on geometry. During the medieval and modern eras, with the advent of algebra, square roots became more widespread. In 1450 AD, Regiomontanus invented a symbol, written as ‘R’ to denote square roots. Years later, in 1525, Christoph Rudolf introduced the now-familiar symbol √ for square root. Today, square roots are used in countless real-life applications such as calculating dimensions, forces, distances, and so on.
The square root has several properties that make it both unique and easy to understand. Let us uncover these properties:
The square root of a number can be classified into two types: perfect squares and non-perfect squares. Let us now understand more about the classifications.
A perfect square is said to be an integer that is the square of another integer. In simple terms, it is a number resulting from squaring another number. As an example, 5 × 5 results 25. Here, 25 is a perfect square and it is the product of 5. Also, 5 is an integer.
Some examples of perfect squares are:
1 = 1 × 1 = 12
4 = 2 × 2 = 22
9 = 3 × 3 = 32
16 = 4 × 4 = 42
25 = 5 × 5 = 52
36 = 6 × 6 = 62
These examples clearly illustrate the concept of a perfect square. A perfect square is the result that is obtained by multiplying an integer by itself.
A non-perfect square root is not an integer. It will be an irrational number instead. It cannot be expressed as a simple fraction. They are numbers that do not result from multiplying an integer by itself. Its decimals never stop.
For example, the square root of 2 is:
√2 ≈ 1.414
The square root of 2, which is 1.414 is an irrational number. Non-perfect squares have square roots whose values are not whole numbers. A few examples of non-perfect squares are:
√2 ≈ 1.414
√3 ≈ 1.732
√5 ≈ 2.236
√8 ≈ 2.828
In advanced mathematics, non-perfect squares are widely used, where approximate values are required.
Let us dive into the world of square roots and understand how it works. It has countless applications in everyday life. Square roots help students to solve academic and real-world problems. Some key benefits of mastering square roots by students are:
In advanced mathematics, square roots are a fundamental element. In algebra, geometry, and calculus they play a significant role in solving quadratic equations and comprehending logarithms and exponents.
Students can apply square roots in different areas such as architecture, construction, engineering, and physics. It is important in calculating distances, forces, and intensity of energies. Also, it plays a crucial role in financial contexts, such as in determining compound interests and analyzing the risks in financial properties.
Understanding square roots helps students enhance their critical thinking and analytical skills. It boosts their confidence in solving challenging math problems and strengthens their confidence in the subject. This foundation enables them to focus on future studies that heavily rely on square roots and mathematics.
There are different methods we use to calculate square roots of a given number. Some of the methods we use are given below:
While using the prime factorization method for square root calculation, a number is represented as the product of its prime factors. Remember that this method is only applicable to perfect square numbers. To find the square root of a perfect square, the prime factorization is the most commonly used method. We need to follow the steps mentioned below to calculate the square root of a number:
Step 1: Break down a number into its prime factors. Start with the smallest prime number, which is 2.
Step 2: Group pairs of factors where both factors in each pair are the same.
Step 3: Take one factor from each pair.
Step 4: Multiply the factors. The square root of a given number is the product of their factors.
For instance, take a look at this example of the square root of 144:
144 = 2 × 2 × 2 × 2 × 3 × 3
(2 × 2) × (2 × 2) × (3 × 3)
(2 × 2 × 3)2
144 = 12 × 2
√144 = 12
The square root of 144 is ±12.
To find the square root of an imperfect number we can apply the long division method. In this method large numbers will be broken down into small parts. The several steps of this method are:
Step 1: Break a number into pairs of two digits from right to left.
Step 2: Find the greatest number whose square is smaller or equal to the first digit or pair.
Step 3: Now we can subtract the square of that number from the pair. After that, we can drop the next pair of numbers.
Step 4: Double the number you found in step 2, and you get the new divisor.
Step 5: When we reach the required level, we can stop the steps.
For example, we can find the square root of 20:
Step 1: We have one pair of digits for 20.
Step 2: 4 is the largest number whose square is smaller than or equal to 20, because (4 × 4 = 16). The first digit of the square root of 20 is 4.
Step 3: Subtract 16 from 20.
20 - 16 = 4
Now, we have to drag down the next pair. But here, there is no other pair. So add a zero and make it 400.
Step 4: Double the digit 4 and we get 8 as the result.
Step 5: Next, we find the largest number, whose square is less than or equal to 400, when we multiply by the new number and add to 8. Here, 5 is the next digit.
8 × 5 = 40
40 × 5 = 200
Step 6: Subtract 200 from 400 which gives 200.
Now, we know the approximate square root of 20 is ±4.5.
This is a simple and interesting way to calculate the square roots of any given number. Finding the symbol of square root √ in the calculator is the foremost thing you should remember. Just enter your number then press the square root symbol √. The calculator will display the square root.
Some tips and tricks will help students to understand more about the square roots. Also, it makes the calculations easier. After applying these tips and tricks, kids can solve difficult mathematical problems and equations.
Primarily, students should study the fundamentals of square roots. It is a value that, when we multiply by itself, we get the original number.
Another useful trick is to by heart the perfect squares of numbers from 1 to 15. For example, we know that, 22 = 4 and 32 = 9. Memorizing these will help students quickly identify perfect squares during calculation.
For non-perfect squares, estimate the nearest perfect squares. For instance, to identify the √20, find the closest perfect squares, they are √16(4) and √25(5).
To simplify the square roots of any given number, break the number into its prime factors. It will help the students to solve problems more easily and faster.
If the given number is large, kids can use the division method to find the square root. By breaking a large number into smaller parts, the calculations become much easier.
While finding the square roots of any number, the calculations might seem tricky. Students may also make several common errors during the solving process. Here are some common mistakes and their solutions. Avoiding these errors will help students make accurate calculations.
Bobby is planting flowers in a square garden with an area of 169 square feet. What is the length of one side of the garden?
The length of one side of the garden is 13 feet
To find the length of one side, we need to calculate the square root of 169.
The formula to calculate the area is:
Area = side × side. It is a square-shaped garden, so all four sides are equal. Hence,
Side = √Area
Side = √169
Now, we have to find the √169.
13 × 13 = 169
Therefore, Side = 13
Find the value of ‘x’ in the equation: √(x+2) = 4
x = 2 or x = -6.
To find the value of ‘x’, in √(x+2) = 4
we need to square both sides:
√(x+2) = 4
x+2 = 4
x= 4 - 2
Thus, x = 2 or x = -6.
What is the square root of the sum of 10 and 6?
The square root of the sum of 10 and 6 is √16, which equals 4.
First, we have to calculate the sum of 10 and 6:
10 + 6 = 16
Then, find the square root of 16:
√16 = 4
The square root of the sum of 10 and 6 is √16, which equals 4.
Joyal is building a square path in his garden. If the area of the path is 100 square feet, what is the length of one side of the path?
To calculate the side length, find the square root of 100.
√100 = 10
The square root of 100 is 10. Therefore, the length of one side of the path is 10 feet.
What is the square root of the product of 18 and 2?
The square root of the product of 18 and 2 is √36, which equals 6.
First we need to multiply the given numbers:
18 × 2 = 36
Now, find the square root of 36:
√36 = 6
The square root of the product of 18 and 2 is √36, which equals 6.