Square Root Calculator

What is the meaning of the symbol of the square root?

The meaning of the square root symbol \(√\) is a positive number whose square root is x.
Example:
\( \sqrt{9} = 3\)
\( \sqrt{9} = \pm 3\)
\( \pm \sqrt{9} = \pm 3\)
\( -\sqrt{9} = -3\)
\( f(x) = \sqrt{x}\)

\(x^2 = 9\)
\(\pm \sqrt{x^2} = \pm \sqrt{9}\)
\(x = \pm 3\)

Let \(\sqrt{2378} \)
max \((\sqrt{2378})\)
\(+\sqrt{2378} \) and \(-\sqrt{2378} \) both roots

Principal square root

Principal square root is a positive square root of a number it can be rational or irrational.

Rational numbers square root

Let’s take \(3^2\) and \((-3)^2\).
For \(3^2\), \(3^2 = 3 ✕ 3 = 9 = \sqrt{9}\)
For \((-3^)2\), \((-3)^2 = (-3) ✕ (-3) = 9 = \sqrt{9}\)

Let’s take \(4^2\) and \((-4)^2\).
For \(4^2\), \(4^2 = 4 ✕ 4 = 16 = \sqrt{16}\)
For \((-4^)2\), \((-4)^2 = (-4) ✕ (-4) = 16 = \sqrt{16}\)

Perfect Squares

Product of multiplying equal numbers. Examples of perfect square:

1- \( \sqrt{64} = 8 \)
2- \( \sqrt{121} = 11 \)
3- \( \sqrt{0.04} = 0.2 \)
4- \( \sqrt{0.49} = 0.7 \)
5- \( \sqrt{\frac{9}{25}} = \frac{3}{5} \)

Square Root of Irrational Numbers

Irrational numbers are,
1- Non perfect square numbers
2- Non repeating
3- Non terminating decimals

Example:

\( \sqrt{2} = 1.414…. \)
\( \sqrt{5} = 2.236…. \)

Principal Square Root of Non-Negative Number

\( x^2 = 4 \)
\( x = 2 => 2^2 = 4 \)
\( x = -2 => (-2)^2 = 4 \)
\( \sqrt{4} = 2\) or \( -2 \) This notation refers specifically to the principal square root of 4 so, principal square root of non-negative number is non negative therefore \( \sqrt{4} = 2 \).

Significance of Signs in Square Root

\( x^2 = 4 \)
\( x = 2 \)

Here one solution is missing.
\( x^2 = 4 \)
\( x^2 – 4 = 0 \)
\( x = 2 \) and \( x = -2 \)
\( 2^2 = 4 \) and \( (-2)^2 = 4 \)

Estimating the Square Root of Numbers to the Nearest Hundred

\( \sqrt{56} \)

Sort out the perfect square root between above and below of 56 those are 7 and 8.
\( (7.5)^2 = 56.25 \)
\( (7.4)^2 = 54.76 \)
\( (7.48)^2 = 55.95 \)
So, \( \sqrt{56} = 7.48 \)

Estimating the square root of 5 to the nearest hundred

Step 1: Find the two consecutive perfect squares which \(\sqrt{5}\) lies.

So, square root of \(\sqrt{5}\) lies between 2 and 3.

Step 2: Estimate the square root to the nearest hundred by using number line.

Alternate method for estimating the \(\sqrt{5}\) to the nearest hundred

From 2.31 to 2.235 it is still very close to 5 but 2.236 is the closest and we need to round it to the nearest hundred so, the closest nearest estimated value is 2.24.

Hence, \(\sqrt{5} = 2.24\)

Uses of square root in practical life (Hero’s formula and Pythagoras theorem)

For calculating area of a specific figure by using hero’s formula:

\(area = \sqrt{s(s – a)(s – b)(s – c)}\)

Here “s” is semi perimeter.

\( S = \dfrac{a + b + c}{2} \)

Example:

Consider \( a = 11 \), \( b = 17 \) and \( c = 16 \).

\( S = \dfrac{a + b + c}{2} \)

\( S = \dfrac{11 + 17 + 16}{2} \)

\( S = 22 \)

Applying Hero’s formula for calculating area

\(area = \sqrt{s(s – a)(s – b)(s – c)}\)

\(area = \sqrt{22(22 – 11)(22 – 16)(22 – 17)}\)

\(area = \sqrt{7620}\)

\(area = 85.5206 \) sq units

Find out the length of articles by applying Pythagoras theorem.

A 32 inch TV is to be bought with dimension 16 and 9.How actually big would be the TV length wise and breadth wise?

Let the common multiple to be x
\(a = 16x\) and \(b = 9x\)

By applying Pythagoras Theorem
\(C^2 = A^2 + B^2\)
\(32^2 = (16x)^2 + (9x)^2\)
\(x = 1.743 \) inch

So, length and breadth of TV is
\(a = 16x = 16(1.74) = 27.838 \) inch
\(b = 9x = 9(1.74) = 15.687\) inch

How to simplify square root?

1- \(\sqrt{49} = 7\)
2- \(\sqrt{-25} = \sqrt{25} \sqrt{-1} = 5i\)
3- \(-\sqrt{81} = -(9) = -9\)
4- \(-\sqrt{-64} = -\sqrt{64} \sqrt{-1} = -(8)i = -8i\)
5- \(\sqrt{18} = \sqrt{9} \sqrt{2} = 3\sqrt{2}\)
6- \(\sqrt{75} = \sqrt{25} \sqrt{3} = 5\sqrt{3}\)
7- \(4\sqrt{98} = 4\sqrt{49} \sqrt{2} = 4(7)\sqrt{2} = 28\sqrt{2}\)

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