Square of a number || Perfect Square of a number || Square Root of a positive real number || Properties of Square Root

Square of a number

• A number's square is the product of the number and the number itself.
• A number whose power is $2$.
• The square of $x$ is $(x$ $\times x)$ denoted by $x^2$.

Examples:

$1^2=$ $1\times 1$ $=1$             $(-1)^2$ $=(-1)(-1)$ $=1$

$2^2=$ $2\times 2$ $=4$             $(-2)^2$ $=(-2)(-2)$ $=4$

$3^2$ $=3\times 3$ $=9$             $(-3)^2$ $=(-3)(-3)$ $=-9$

$4^2$ $=4\times 4$ $=16$           $(-4)^2$ $=(-4)(-4)$ $=16$

$5^2$ $=5\times 5$ $=25$           $(-5)^2$ $=(-5)(-5)$ $=-25$

$6^2$ $=6\times 6$ $=36$           $(-6)^2$ $=(-6)(-6)$ $=26$

$7^2$ $=7\times 7$ $=9$             $(-7)^2$ $=(-7)(-7)$ $=-49$

$8^2$ $=8\times 8 $=64$           $(-8)^2$ $=(-8)(-8)$ $=64$

$9^2$ $=9$ $\times 9$ $=81$           $(-9)^2$ $=(-9)(-9)$ $=-81$

$3^2$ $=3$ $\times 3$ $=9$              $(-3)^2$ $=(-3)(-3)$ $=9$

$\left( \frac{1}{2} \right)^2$ $=\left( \frac{1}{2} \right)$ $\left( \frac{1}{2} \right)$ $=\frac{1}{4}$      $\left(- \frac{1}{2} \right)^2$ $=\left(- \frac{1}{2} \right)$ $\left(- \frac{1}{2} \right)$ $=\frac{1}{4}$

                  $\left( \frac{3}{2} \right)^2$ $=\left( \frac{3}{2} \right)$ $\left( \frac{3}{2} \right)$ $=\frac{9}{4}$          $\left(-\frac{3}{2} \right)^2$ $=\left( -\frac{3}{2} \right)$ $\left(- \frac{3}{2} \right)$ $=\frac{9}{4}$

Perfect Square of a number

• A natural number that is a square of another number is called a perfect square of that number.

i.e.

$9$ $=3^2$,     $9$ is a perfect square of $3$

$16$ $=4^2$,    $16$ is a perfect square of $4$

$1$ $=1^2$,      $1$ is a perfect square of $1$

$64$ $=8^2$,    $64$ is a perfect square of $8$

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$n$ $\times n$ $=n^2$,      $(n$ $\times n)$ is a perfect square of $n$

                  

We can find the perfect square of rational numbers
$\frac{9}{4}$ $=\left( \frac{3}{2} \right)^2$,      $\frac{9}{4}$ is perfect square of $\frac{3}{2}$

 

We can not find perfect squares of irrational numbers
$\left( \sqrt 2 \right)^2$ $=2$      is a square but not a perfect square

$\left( \pi \right)^2$ $=\pi^2$       is not a perfect square

 

Square Root of a positive real number

• The square root of a positive real number is a number that when multiplied by itself gives the same number.
• A number's square root is one of its two equal factors.
• A radical sign “$\sqrt{ }$”, is the symbol used to indicate a square root.
• It is also called “principal square root”.
  
  

In general for a positive real number $q$ the square root of $q$ will $\sqrt q$.

In the notation $\sqrt q$, $“\sqrt{}”$ is called the “radical sign” and $q$ is called the “radicand”.

Hence, $\sqrt q$ stands for a positive real number $x$ for which

$q=$ $x^2$

 

Properties of Square Root:

$i)$  $\sqrt a$ $=x$ $\implies a$ $=x^2$,    $a$ $\ge 0$

Proof:

$a^{\frac{1}{2}}$ $=x$

Squaring on b.sides

$a$ $=x^2$

 

$ii)$   $\sqrt a$ $\sqrt a$ $=a$,     $a$ $\ge 0$

Proof:

$a^{1/2}\times a^{1/2}$ $=a^{1/2+1/2}$ $=a^1$ $=a$

$\sqrt a$ $\sqrt a$ $=a$

 

$iii)$   $\sqrt a$ $\sqrt b$ $=\sqrt{ab}$,   $a,b$ $\ge0$

Proof:

$\sqrt a^{1/2}$ $\sqrt b^{1/2}$ $=(ab)^{1/2}$ $=\sqrt {ab}$

$\sqrt a$ $\sqrt b$ $=\sqrt {ab}$

 

$iv)$   $\frac{a}{a}$ $=1$,   $a$ $\gt 0$

Proof:

$\frac{\sqrt a}{\sqrt a}$ $=\frac{a^{1/2}}{a^{1/2}}$ $=a^{\frac{1}{2}$ $-\frac{1}{2}}$ $=a^0$ $=1$

$\frac{\sqrt a}{\sqrt a}$ $=1$

 

$v)$   $\frac{a}{\sqrt a}$ $=\sqrt a$,    $a$ $\gt 0$

Proof:

$\frac{a}{\sqrt a}$ $=\frac{a}{a^{1/2}}$ $=a^{1-\frac{1}{2}}$ $=a^{\frac{1}{2}}$ $=\sqrt a$

$\frac{a}{\sqrt a}$ $=\sqrt a$

 

$vi)$   $\frac{\sqrt a}{a}$ $=\frac{1}{\sqrt a}$,      $a\gt 0$

Proof:

$\frac{\sqrt a}{a}$ $=\frac{a^{\frac{1}{2}-1}}{a}$ $=a^{\frac{1}{2}-1}$ $=a^{-\frac{1}{2}}$ $=\frac{1}{\sqrt a}$

$\frac{\sqrt a}{a}$ $=\frac{1}{\sqrt a}$