overview

In this page, square of a number and square root of a number are revised. The long division method to find the square root is explained.

square

We studied in exponents that ${3}^{2}=3\times 3=9$.

An exponent of power $2$ is called *square* of the number.

eg: Square of $7$ is ${7}^{2}=49$.

The word "square" means : the 2D shape of equal sides and $90}^{\circ$ angles" between the sides. The exponent square is representative of "area of the shape square, side to the power $2$".

root

We studied in the exponents that ${16}^{\frac{1}{2}}=\sqrt[2]{16}=4$".

An exponent of power $\frac{1}{2}$ or the $2$nd root is called *square root* of the number.

eg: Square root of $64$ is ${64}^{\frac{1}{2}}=\sqrt[2]{64}=8$.

The power of the root need not be mentioned for square roots

$\sqrt[2]{64}=\sqrt{64}$.

examples

Square of $4$ is $4}^{2$.

Square root of $9$ is $\sqrt[2]{9}={9}^{\frac{1}{2}}$.

$\sqrt{100}={100}^{\frac{1}{2}}=10$".

the symbol $\sqrt{}$ is the square root.

definitions

**Square of a number** : A number multiplied by itself is the square of the number.

eg: ${6}^{2}=6\times 6=36$

**Square Root of a number** : Square root of a value is the number whose square is the given value.

eg: $\sqrt{36}=6$ as ${6}^{2}=36$.

factorize

The factors of $26$ are "$1,2,13,26$"

The factors of $28$ are "$1,2,4,7,14,28$"

The prime factorization of $264$ is $2\times 2\times 2\times 3\times 11$.

A procedure is illustrated in the figure.

finding square roots

Consider finding the result of $\sqrt{3600}$.

Square root is a form of root. In roots, we learned to perform prime factorization to find the root.

$\sqrt{3600}$

$=\sqrt{2\times 2\times 2\times 2\times 3\times 3\times 5\times 5}$

$=2\times 2\times 3\times 5$

$=60$

**Finding Square Roots** : To find square root of a number, express the number in prime factors and group the factors.

eg: $\sqrt{100}$ $=\sqrt{2\times 2\times 5\times 5}$ $=2\times 5=10$

Note: This method is suitable for finding square roots resulting in integers.

example

What is the $\sqrt{8}$

The answer is "$2\sqrt{2}$".

Using the prime factorization method

$\sqrt{8}$

$=\sqrt{2\times 2\times 2}$

$=2\times \sqrt{2}$

The prime factorization method is suitable for square roots resulting in integer values.

a procedure

Square root of a number can be given as

$x}^{2}={(10a+b)}^{2$

$x}^{2}=100{a}^{2}+\left(20ab\right)+{b}^{2$

Note:

$b}^{2$ is a $2$ digit number with tens-units places

$100{a}^{2}$ is a number that has $00$ at tens-units places

This understanding gives a method to eliminate $b$ and look at $a$ to choose the highest digit of the square root.

$x}^{2}=\textcolor[rgb]{}{100{a}^{2}}+\textcolor[rgb]{}{(20a+b)\times b$

$x}^{2}\textcolor[rgb]{}{-100{a}^{2}}=\textcolor[rgb]{}{(20a+b)\times b$

The rearranged one with $(20a+b)$ gives the method to multiply $a$ by $2$ (which is $2a$) and append a value $b$, which is $20a+b$. Then multiply $b$ to $20a+b$.

*step 1 *

${x}^{2}\textcolor[rgb]{}{-{a}^{2}\phantom{\rule{1ex}{0ex}}\text{at 100s place}}=y$

*step 2 *

$y-\textcolor[rgb]{}{\left(2a\phantom{\rule{1ex}{0ex}}\text{joined with}\phantom{\rule{1ex}{0ex}}b\right)\times b}$

In this process, the choice of $a$ and $b$ make the square root $x=10a+b=a\phantom{\rule{1ex}{0ex}}\text{joined with}\phantom{\rule{1ex}{0ex}}b$

The above process is explained for 2 digit square root and is easily extended for higher number of digits.

example using the procedure

Consider $\sqrt{529}$

$529$

$={23}^{2}$

$={(10\times 2+3)}^{2}$

$={(10\times 2)}^{2}+2\times 10\times 2\times 3+{3}^{2}$

$=\textcolor[rgb]{}{100\times {2}^{2}}+\textcolor[rgb]{}{(2\times 2\times 10+3)\times 3}$

$=\textcolor[rgb]{}{{2}^{2}\phantom{\rule{1ex}{0ex}}\text{100s place}\phantom{\rule{1ex}{0ex}}}+\textcolor[rgb]{}{(2\times 2\phantom{\rule{1ex}{0ex}}\text{joined}\phantom{\rule{1ex}{0ex}}3=43)\times 3}$

*The above steps is used in reverse when the square root is not known *

$\sqrt{529}$

$=\textcolor[rgb]{}{2}\phantom{\rule{1ex}{0ex}}\text{at 10s place}\phantom{\rule{1ex}{0ex}}+\textcolor[rgb]{}{3}\phantom{\rule{1ex}{0ex}}\text{at units place}$

$=23$

This procedure is illustrated in the figure.

$\sqrt{69169}=263$

summary

**Procedure to finding Square Root of a number** : Long division method is illustrated in the figure.

Outline

The outline of material to learn "Exponents" is as follows.
Note: * click here for detailed outline of Exponents s *

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