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 $8$
.
.
.
$n$ $\times n$ $=n^2$, $(n$ $\times n)$ 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:
$\frac{\sqrt a}{a}$ $=\frac{a^{\frac{1}{2}-1}}{a}$ $=a^{\frac{1}{2}-1}$ $=a^{-\frac{1}{2}}$ $=\frac{1}{\sqrt a}$
$\frac{\sqrt a}{a}$ $=\frac{1}{\sqrt a}$
0 Comments