Table of Contents
What is square root? - Square Root of 7
The term "square root" refers to a
mathematical operation that determines the value that, when multiplied by itself,
gives the original number. In other words, the square root of a number is a value that, when squared
(multiplied by
itself), equals the original number. The square root symbol is represented by a radical (√).
What is a perfect square?
- Square Root of 7
A perfect square is a number that can be expressed as
the square of an integer. In other words, it is a number that has
an exact integer square root. When you take the square root of a perfect square, you get a whole number as
the result.
For example, 9 is a perfect square because it can be expressed as 3 × 3, where 3 is an
integer. Similarly, 16 is a
perfect square because it can be expressed as 4 × 4.
Some common examples of perfect squares are 1, 4, 9, 16, 25, 36, 49, and so on.
How do we calculate square root? - Square
Root of 7
The square root of a number can be calculated
using various methods. Here are two commonly used methods:
1. Prime Factorization Method: This
method involves finding the prime factors of the number and grouping them into
pairs. Each pair consists of the same prime factor. The product of these pairs gives the square root of
the original number.
For example, to calculate the
square root of 36 using prime factorization:
The prime factors of 36 are 2, 2, 3, 3.
Grouping them into pairs: (2, 2), (3, 3).
Taking the product of each pair: 2 × 2 × 3 × 3 = 6 × 6 = 36.
So, the square root of 36 is 6.
2. Long Division Method: This method is
similar to long division, but it involves finding the square root digit by
digit.
For example, to calculate the
square root of 36 using prime factorization:
Start by separating the digits into pairs from the right: 8 and 1.
Find the largest digit 'x' such that 'x' multiplied by itself is less than or
equal to 8 (the leftmost pair). In this
case, 'x' is 2 because 2 × 2 = 4.
Write 2 as the first digit of the square root.
Subtract the square of 2 from 8: 8 - 4 = 4.
Bring down the next pair (1) to the right of 4.
Double the current digit (2) and guess a digit for the next place value. Try 4
(2 multiplied by 2) as a guess.
Multiply the new number (24) by the guessed digit (4): 24 × 4 = 96.
Find the largest single-digit number 'x' that satisfies the condition '2x
multiplied by x' is less than or equal to 96.
In this case, 'x' is 6 because 26 × 6 = 156.
Write 6 as the next digit of the square root.
Subtract the product (156) from the result (196): 196 - 156 = 40.
Since there are no more pairs, the square root of 81 is 26.
What is Prime Factorization Method? - Square Root of 7
The Prime Factorization Method is a technique
used to determine the prime factors of a number. It involves breaking down
the given number into its constituent prime factors, which are prime numbers that can divide the original
number evenly.
Here's an overview of the Prime Factorization Method:
- Start with the number you want to find the prime
factors of.
- Begin by dividing the number by the smallest
prime number, which is 2. If the number is divisible by 2, divide it
repeatedly until it is no longer divisible. Note down each time you divide by 2.
- Move on to the next prime number, which is 3. If
the number is divisible by 3, divide it repeatedly until it is no
longer divisible. Note down each time you divide by 3.
Continue this process with successive prime numbers
(5, 7, 11, and so on) until the number becomes 1.
The list of all the prime numbers that you noted
down during the process is the prime factorization of the original
number.
The prime factorization method helps identify all the prime factors of a given
number, which can be useful for various
purposes such as simplifying fractions, finding the greatest common divisor, or calculating square roots.
What is Long Division Method? - Square
Root of 7
The Long Division Method is a mathematical
technique used to perform division when the dividend (the number being
divided) and divisor (the number dividing the dividend) are relatively large or when an exact quotient is
required. This
method involves a step-by-step process of dividing the dividend by the divisor, digit by digit, to find the
quotient.
Here's a step-by-step explanation
of the Long Division Method:
- Write the dividend (the number being divided) on
the left, and the divisor (the number dividing the dividend) on the
outside of the division symbol (÷).
- Start with the leftmost digit of the dividend
and divide it by the divisor. Write the quotient (the result of the
division) on top of the division symbol, directly above the dividend digit.
- Multiply the quotient by the divisor and write
the result below the dividend, aligned with the corresponding digits.
- Subtract the product obtained in step 3 from the
digits directly above it. Write the difference below the line.
- Bring down the next digit of the dividend and
write it next to the difference obtained in the previous step.
- Repeat steps 2 to 5 until all the digits in the
dividend have been processed.
- Once the last digit of the dividend has been
brought down and processed, the division is complete.
The final result is the quotient obtained from the division.
How to Estimate the Square Root? - Square
Root of 7
Estimating the
square root of a number involves finding an approximate value that is close to the actual square root.
Here's a method to estimate the square root:
- Identify the nearest perfect square below the
given number. This will serve as a reference point for estimation.
- Determine the difference between the given
number and the nearest perfect square found in step 1.
- Divide the difference obtained in step 2 by
twice the reference point (the square root of the perfect square found in
step 1).
- Add the result obtained in step 3 to the
reference point to get an estimated value of the square root.
Fun Facts - Square Root of 7
The symbol for the square root (√) was first introduced by the ancient Greek mathematician Heron of Alexandria. He used
a special symbol to represent square roots in his work on geometry and engineering, which dates back to around the 1st
century AD.
However, the concept of square roots predates Heron of Alexandria by several centuries. The Babylonians, who lived
around 2000 BC, had a basic understanding of square roots and used approximation methods to calculate them. The concept
of square roots also appeared in the works of ancient Indian mathematicians, such as Aryabhata in the 5th century AD.
The discovery and understanding of square roots have played a significant role in the development of mathematics and
have applications in various fields, including engineering, physics, computer science, and cryptography.
FAQ - Square Root of 7
What is the meaning of √?
Square root
What is this √ 2?
The expression √2 represents the square root of 2. It is an irrational
number, meaning it cannot be expressed as a
fraction (ratio) of two integers and its decimal representation goes on infinitely without repeating.
The square root of 2 is approximately equal to 1.41421356
Why is √ 4 = 2 only?
The square root of 4, denoted as √4, is equal to 2. This is because when you
square 2 (multiply it by itself), the
result is 4.
How to solve √ 3?
The square root of 3, denoted as √3, is approximately equal to 1.732 when
rounded to three decimal places. The long
division method can be used to calculate the square root of 3, but it is a non-repeating, irrational number.
Its exact
value extends infinitely without a pattern. However, for practical purposes, the value of √3 is often
approximated as
1.732 to simplify calculations.
What is the square of √ 1?
1
Why √2 is irrational?
The irrationality of the square root of 2 was discovered by the ancient
Greeks. They observed that the diagonal of a
square with sides measuring 1 unit cannot be expressed as a rational number. Using the Pythagorean Theorem,
they
determined that the length of the diagonal is equal to the square root of 2. Therefore, it was concluded
that the square
root of 2 is an irrational number, meaning it cannot be expressed as a fraction or ratio of two integers.
The decimal
representation of the square root of 2 goes on infinitely without repeating or terminating.
Can I take the square root of
0?
Yes, the answer is 0
Is Infinity a square root?
In mathematics, infinity is not treated as a number, and the concept of
taking the square root of infinity does not have
a well-defined meaning.
Is root a real number?
The square root of a non-negative real number is a real number, while the
square root of a negative real number is a
complex number involving the imaginary unit "i."
Is 4 a square root?
No, 4 is not a square root. 4 is a perfect square, meaning that it is the
result of squaring a whole number. In this
case, 4 is the square of 2, because 2 multiplied by 2 equals 4.
