Squire Root Calculator
Calculating square root
A number produces a particular number when multiplied by itself. 7 x7 =49. We say 7 is a square root of 49.
The easiest way to calculate a square root is by using a calculator. What is the formula to calculate the square root of a random number? There is a formula to express the square root of a number. The formula is √y = y½
If a random number has an exponent of ½, then a square root is needed to be found for that random number.
Square Root Calculator Formula
Square Root Formula
The square root of a number \( x \) can be represented as:
\[ \sqrt{x} \]
Methods to Calculate Square Roots
1. Using the Exponentiation Formula
You can also express the square root as an exponent:
\[ \sqrt{x} = x^{0.5} \]
2. Newton's Method
Newton's method is an iterative numerical method to approximate square roots. The formula for updating the approximation is:
\[ x_{n+1} = \frac{1}{2} \left( x_n + \frac{x}{x_n} \right) \]
Here, \( x_n \) is the current approximation, and \( x_{n+1} \) is the next approximation. Repeat this until \( x_n \) is sufficiently close to \( x_{n+1} \).
3. Using Built-in Functions
In many programming languages, there are built-in functions to compute square roots. For example:
- In Python:
math.sqrt(x) - In Java:
Math.sqrt(x)
Example
To find the square root of 16:
Using the formula: \(\sqrt{16} = 4\)
Using Newton's method, start with an initial guess \( x_0 = 4 \):
\[ x_1 = \frac{1}{2} \left( 4 + \frac{16}{4} \right) = \frac{1}{2} (4 + 4) = 4 \]
In the past, there was no calculator. Teachers and students had to find the square root by themselves. It was their only option.
There are multiple methods to find the square root of a number by hand.
Method 1 uses prime factorization. In this method, numbers must be divided into perfect square factors. A number factor is any set of numbers that multiply to make a number.
Example: √36 = √(4*9)=√4 * √9 = 2*3= 6
Square Root Examples
Example 1
Calculate the square root of \( 16 \).
Solution: \( \sqrt{16} = 4 \)
Example 2
Find the square root of \( 25 \).
Solution: \( \sqrt{25} = 5 \)
Example 3
What is the square root of \( 36 \)?
Solution: \( \sqrt{36} = 6 \)
Example 4
Calculate the square root of \( 49 \).
Solution: \( \sqrt{49} = 7 \)
Example 5
Find the square root of \( 64 \).
Solution: \( \sqrt{64} = 8 \)
Example 6
What is the square root of \( 81 \)?
Solution: \( \sqrt{81} = 9 \)
Example 7
Calculate the square root of \( 100 \).
Solution: \( \sqrt{100} = 10 \)
Example 8
Find the square root of \( 121 \).
Solution: \( \sqrt{121} = 11 \)
Example 9
What is the square root of \( 144 \)?
Solution: \( \sqrt{144} = 12 \)
Example 10
Calculate the square root of \( 169 \).
Solution: \( \sqrt{169} = 13 \)
Example 11
Find the square root of \( 196 \).
Solution: \( \sqrt{196} = 14 \)
Example 12
What is the square root of \( 225 \)?
Solution: \( \sqrt{225} = 15 \)
Example 13
Calculate the square root of \( 256 \).
Solution: \( \sqrt{256} = 16 \)
Example 14
Find the square root of \( 289 \).
Solution: \( \sqrt{289} = 17 \)
Example 15
What is the square root of \( 324 \)?
Solution: \( \sqrt{324} = 18 \)
Example 16
Calculate the square root of \( 361 \).
Solution: \( \sqrt{361} = 19 \)
Example 17
Find the square root of \( 400 \).
Solution: \( \sqrt{400} = 20 \)
Example 18
What is the square root of \( 441 \)?
Solution: \( \sqrt{441} = 21 \)
Example 19
Calculate the square root of \( 484 \).
Solution: \( \sqrt{484} = 22 \)
Example 20
Find the square root of \( 529 \).
Solution: \( \sqrt{529} = 23 \)
Square Root Examples Involving Points
Example 1
Calculate the distance between points \( A(1, 2) \) and \( B(4, 6) \).
Solution:
The distance \( d \) is given by:
\[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2}
= \sqrt{9 + 16} = \sqrt{25} = 5 \]
Example 2
Find the distance between points \( A(-3, -1) \) and \( B(1, 3) \).
Solution:
\[ d = \sqrt{(1 - (-3))^2 + (3 - (-1))^2} = \sqrt{(1 + 3)^2 + (3 + 1)^2}
= \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
Example 3
Calculate the distance between points \( A(0, 0) \) and \( B(5, 5) \).
Solution:
\[ d = \sqrt{(5 - 0)^2 + (5 - 0)^2} = \sqrt{5^2 + 5^2}
= \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]
Example 4
Find the distance between points \( A(2, 3) \) and \( B(6, 7) \).
Solution:
\[ d = \sqrt{(6 - 2)^2 + (7 - 3)^2} = \sqrt{4^2 + 4^2}
= \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
Example 5
Calculate the distance between points \( A(1, -2) \) and \( B(4, 2) \).
Solution:
\[ d = \sqrt{(4 - 1)^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2}
= \sqrt{9 + 16} = \sqrt{25} = 5 \]
Example 6
Find the distance between points \( A(-1, -1) \) and \( B(2, 3) \).
Solution:
\[ d = \sqrt{(2 - (-1))^2 + (3 - (-1))^2} = \sqrt{3^2 + 4^2}
= \sqrt{9 + 16} = \sqrt{25} = 5 \]
Example 7
Calculate the distance between points \( A(3, 7) \) and \( B(3, 1) \).
Solution:
\[ d = \sqrt{(3 - 3)^2 + (1 - 7)^2} = \sqrt{0^2 + (-6)^2}
= \sqrt{0 + 36} = \sqrt{36} = 6 \]
Example 8
Find the distance between points \( A(4, 4) \) and \( B(0, 0) \).
Solution:
\[ d = \sqrt{(0 - 4)^2 + (0 - 4)^2} = \sqrt{(-4)^2 + (-4)^2}
= \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
Example 9
Calculate the distance between points \( A(7, 8) \) and \( B(2, 3) \).
Solution:
\[ d = \sqrt{(2 - 7)^2 + (3 - 8)^2} = \sqrt{(-5)^2 + (-5)^2}
= \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]
Example 10
Find the distance between points \( A(-5, -3) \) and \( B(1, 1) \).
Solution:
\[ d = \sqrt{(1 - (-5))^2 + (1 - (-3))^2} = \sqrt{(1 + 5)^2 + (1 + 3)^2} = \sqrt{6^2 + 4^2}
= \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]
Example 11
Calculate the distance between points \( A(8, 8) \) and \( B(0, 0) \).
Solution:
\[ d = \sqrt{(0 - 8)^2 + (0 - 8)^2} = \sqrt{(-8)^2 + (-8)^2}
= \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \]
Example 12
Find the distance between points \( A(6, 2) \) and \( B(6, 10) \).
Solution:
\[ d = \sqrt{(6 - 6)^2 + (10 - 2)^2} = \sqrt{0^2 + 8^2}
= \sqrt{0 + 64} = \sqrt{64} = 8 \]
Example 13
Calculate the distance between points \( A(9, 3) \) and \( B(12, 7) \).
Solution:
\[ d = \sqrt{(12 - 9)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Example 14
Find the distance between points \( A(1, 4) \) and \( B(4, 4) \).
Solution:
\[ d = \sqrt{(4 - 1)^2 + (4 - 4)^2} = \sqrt{3^2 + 0^2} = \sqrt{9 + 0} = \sqrt{9} = 3 \]
Example 15
Calculate the distance between points \( A(-1, -2) \) and \( B(1, 3) \).
Solution:
\[ d = \sqrt{(1 - (-1))^2 + (3 - (-2))^2}
= \sqrt{(1 + 1)^2 + (3 + 2)^2} = \sqrt{2^2 + 5^2}
= \sqrt{4 + 25} = \sqrt{29} \]
Example 16
Find the distance between points \( A(-5, 4) \) and \( B(-1, -2) \).
Solution:
\[ d = \sqrt{(-1 - (-5))^2 + (-2 - 4)^2} = \sqrt{(4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \]
Example 17
Calculate the distance between points \( A(2, -1) \) and \( B(2, 4) \).
Solution:
\[ d = \sqrt{(2 - 2)^2 + (4 - (-1))^2} = \sqrt{0 + 5^2} = \sqrt{25} = 5 \]
Example 18
Find the distance between points \( A(1, 1) \) and \( B(7, 4) \).
Solution:
\[ d = \sqrt{(7 - 1)^2 + (4 - 1)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \]
Example 19
Calculate the distance between points \( A(3, 3) \) and \( B(-3, -3) \).
Solution:
\[ d = \sqrt{(-3 - 3)^2 + (-3 - 3)^2} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} \]
Example 20
Find the distance between points \( A(0, 5) \) and \( B(0, -5) \).
Solution:
\[ d = \sqrt{(0 - 0)^2 + (-5 - 5)^2} = \sqrt{0 + (-10)^2} = \sqrt{100} = 10 \]