Indices, Root and Surds

Laws of the indices

1. a^m× aⁿ=a^m+n

2. $\frac{{{{\rm{a}}^{\rm{m}}}}}{{{{\rm{a}}^{\rm{n}}}}}$=a^m-n

3.a⁰= 1

4. (a^m)ⁿ = a^mn

5.${\rm{\: }}\frac{{{{\rm{a}}^{\rm{m}}}}}{{{{\rm{a}}^{\rm{n}}}}}{\rm{\: }}$⁼${\left( {\frac{{\rm{a}}}{{\rm{b}}}{\rm{\: }}} \right)^{{\rm{m}} - {\rm{n\: }}}}$

6.a^m= $\frac{1}{{{{\rm{a}}^{ - {\rm{m}}}}{\rm{\: }}}}$

7. a^m/n = $\sqrt[{\rm{n}}]{{{{\rm{a}}^{\rm{m}}}}}$

8. If a^m= bⁿthen,a= b^n/m

9. if a^m= bⁿthen, m=n

10. a^m= bⁿthen, a =b

 

Examples 1

$\frac{1}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{a}} - {\rm{b\: }}}} + {{\rm{x}}^{{\rm{c}} - {\rm{b}}}}}}$+ $\frac{1}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{b}} - {\rm{c\: }}}} + {{\rm{x}}^{{\rm{a}} - {\rm{c}}}}}}$ +$\frac{1}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{b}} - {\rm{c\: }}}} + {{\rm{x}}^{{\rm{a}} - {\rm{c}}}}}}$

 

soln

$\frac{1}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{a}} - {\rm{b\: }}}} + {{\rm{x}}^{{\rm{c}} - {\rm{b}}}}}}$+ $\frac{1}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{b}} - {\rm{c\: }}}} + {{\rm{x}}^{{\rm{a}} - {\rm{c}}}}}}$ +$\frac{1}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{b}} - {\rm{c\: }}}} + {{\rm{x}}^{{\rm{a}} - {\rm{c}}}}}}$

= $\frac{{{{\rm{x}}^{\rm{b}}}}}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{a}} - {\rm{b\: }}}} + {{\rm{x}}^{{\rm{c}} - {\rm{b}}}}}}$+ $\frac{{{{\rm{x}}^{\rm{c}}}}}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{b}} - {\rm{c\: }}}} + {{\rm{x}}^{{\rm{a}} - {\rm{c}}}}}}$ +$\frac{{{{\rm{x}}^{\rm{a}}}}}{{{\rm{\: }}1 + {{\rm{x}}^{{\rm{b}} - {\rm{c\: }}}} + {{\rm{x}}^{{\rm{a}} - {\rm{c}}}}}}$

= $\frac{{{{\rm{x}}^{\rm{b}}}}}{{{\rm{\: }}{{\rm{x}}^{\rm{b}}}(1 + {{\rm{x}}^{{\rm{a}} - {\rm{b\: }}}} + {{\rm{x}}^{{\rm{c}} - {\rm{b}}}})}}$+ $\frac{{{{\rm{x}}^{\rm{c}}}}}{{{\rm{\: }}{{\rm{x}}^{\rm{c}}}{\rm{\: }}(1 + {{\rm{x}}^{{\rm{b}} - {\rm{c\: }}}} + {{\rm{x}}^{{\rm{a}} - {\rm{c}}}}){\rm{\: }}}}$ + $\frac{{{{\rm{x}}^{\rm{a}}}}}{{{\rm{\: }}{{\rm{x}}^{\rm{a}}}(1 + {{\rm{x}}^{{\rm{b}} - {\rm{c\: }}}} + {{\rm{x}}^{{\rm{a}} - {\rm{c}}}})}}$

= $\frac{{{{\rm{x}}^{\rm{b}}}}}{{{{\rm{x}}^{\rm{b}}} + {{\rm{x}}^{\rm{a}}} + {{\rm{x}}^{\rm{c}}}}}$ + $\frac{{{{\rm{x}}^{\rm{c}}}}}{{{{\rm{x}}^{\rm{b}}} + {{\rm{x}}^{\rm{a}}} + {{\rm{x}}^{\rm{c}}}}}$ +$\frac{{{{\rm{x}}^{\rm{a}}}}}{{{{\rm{x}}^{\rm{b}}} + {{\rm{x}}^{\rm{a}}} + {{\rm{x}}^{\rm{c}}}}}$

=$\frac{{{{\rm{x}}^{\rm{a}}} + {\rm{\: }}{{\rm{x}}^{\rm{b}}} + {{\rm{x}}^{\rm{c}}}}}{{{{\rm{x}}^{\rm{b}}} + {{\rm{x}}^{\rm{a}}} + {{\rm{x}}^{\rm{c}}}}}$

= 1

 

Examples 2

$\frac{1}{{1 + {{\rm{x}}^{\rm{p}}} + {{\rm{x}}^{ - {\rm{q}}}}}} + \frac{1}{{1 + {{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{ - {\rm{r}}}}}} + \frac{1}{{1 + {{\rm{x}}^{\rm{r}}} + {{\rm{x}}^{ - {\rm{p}}}}}}$

 

Soln

= $\frac{1}{{1 + {{\rm{x}}^{\rm{p}}} + {{\rm{x}}^{ - {\rm{q}}}}}} + \frac{1}{{1 + {{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{ - {\rm{r}}}}}} + \frac{1}{{1 + {{\rm{x}}^{\rm{r}}} + {{\rm{x}}^{ - {\rm{p}}}}}}$

= $\frac{1}{{{{\rm{x}}^{ - {\rm{q}} + {\rm{q}}}} + {{\rm{x}}^{{\rm{q}} - {\rm{r}}}} + {{\rm{x}}^{ - {\rm{b}}}}}} + {\rm{\: }}\frac{1}{{1 + {{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{ - {\rm{r}}}}}}{\rm{\: }} + \frac{1}{{{{\rm{x}}^{ + {\rm{r}} - {\rm{r}}}} + {{\rm{x}}^{\rm{r}}} + {{\rm{x}}^{{\rm{q}} + {\rm{r}}}}}}$              

= $\frac{{{{\rm{x}}^{\rm{b}}}}}{{{{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{ - {\rm{r}}}} + 1}} + \frac{1}{{1 + {{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{\rm{r}}}}}{\rm{\: }} + \frac{1}{{{{\rm{x}}^{ + {\rm{r}} - {\rm{r}}}} + {{\rm{x}}^{\rm{r}}} + {{\rm{x}}^{{\rm{q}} + {\rm{r}}}}}}$

 = $\frac{{{{\rm{x}}^{\rm{q}}}}}{{{{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{ - {\rm{r}}}} + 1}} + \frac{1}{{1 + {{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{ - {\rm{r}}}}}}{\rm{\: }} + \frac{{{{\rm{x}}^{ - {\rm{r}}}}}}{{{{\rm{x}}^{{\rm{rc}}}} + {{\rm{x}}^{\rm{q}}} + {\rm{\: }}1}}$

=-$\frac{{1 + {{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{ - {\rm{r}}}}}}{{1 + {{\rm{x}}^{\rm{q}}} + {{\rm{x}}^{ - {\rm{r}}}}}}{\rm{\: }}$=1

 

Examples 3

4^x-6.2^x+1+32 = 0

Soln

4^x-6.2^x+1+32 = 0

2 ^2x-6.2^x+1+32 = 0

2 ^2x-6.2^x .2+32 = 0

let , 2^x=P

P² -12P +32 = 0

P² - 8P – 4P + 32 =0

P(P -8) – 4 (P -8) = 0

(P -4)(P-8) = 0

Either P= 4 or P = 8

 2^x = 2²,

x = 2

2^x =8

X= 3

 

Examples 4

9^a -10*3^a + 9 = 0

Soln

9^a -10*3^a + 9 = 0

Or , 3^2a – 10.3^a + 9 = 0

Or, let 3^a= x

x² -10x +9 = 0

x² – 9x –x+9 =0

x(x- 9) -1(x- 9)= 0

(x-1)(x-9) =0

Either x= 1 Or 3^a= 1

Or,3^a = 3⁰

Or, a=0

x =9

3^a = 3²

a =2

 

Rational Number 

A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator and the denominator are whole numbers.

Surd

A surd is an irrational number, a number which cannot be expressed as a fraction or as a terminating or recurring decimal. It is left as a square root. It can also be a non-cube number left in cube root form and so on.

A radical equation is the equation containing square root or cube root.

Example 1

Solve

$\sqrt {{{\rm{x}}^2} - 2{\rm{x}}} $-1 = x

soln

$\sqrt {{{\rm{x}}^2} - 2{\rm{x}}} $-1 = x

$\sqrt {{{\rm{x}}^2} - 2{\rm{x}}} $= x + 1

Squaring on the both sides

${{\rm{x}}^2} - 2{\rm{x}}$= (x + 1)²

x²-2x =x²+2x+1

x =$ - \frac{1}{4}$

 

Example 2

$\sqrt {{{\rm{x}}^2} + 7{\rm{x}}} {\rm{\: }} - 1{\rm{\: }} = {\rm{x}}$

a. Soln:

or, $\sqrt {{{\rm{x}}^2} + 7{\rm{x}}} {\rm{\: }} - 1{\rm{\: }} = {\rm{x}}$

or,${\rm{\: }}{\left( {\sqrt {{{\rm{x}}^2} + 7{\rm{x}}} {\rm{\: }}} \right)^2}$ = (x+1)²

or, x²+ 7x=x²+ 2x + 1

or, 5x=1

∴ x = $\frac{1}{5}$

 

Example -3

  $\sqrt {4{{\rm{m}}^2} - 7} {\rm{\: }} + 1{\rm{\: }} = 2{\rm{m}}$

a. Soln:

or, $\sqrt {4{{\rm{m}}^2} - 7} {\rm{\: }} + 1{\rm{\: }} = 2{\rm{m}}$

or, ${\left( {\sqrt {4{{\rm{m}}^2} - 7} } \right)^2}$ =${\left( {2{\rm{m}} + 1} \right)^2}$

or, $4{{\rm{m}}^2} - 7$ =$4{{\rm{m}}^2} + 4{\rm{m\: }} + 1{\rm{\: }}$

or, -8=4m

or, m=-2

 

Example 4

   $\sqrt {{{\rm{x}}^2} + 7{\rm{x}}} {\rm{\: }} - 1 = {\rm{x}}$

a. Soln:

or, $\sqrt {{{\rm{x}}^2} + 7{\rm{x}}} {\rm{\: }} - 1{\rm{\: }} = {\rm{x}}$

or,${\rm{\: }}{\left( {\sqrt {{{\rm{x}}^2} + 7{\rm{x}}} {\rm{\: }}} \right)^2}$ = (x+1)²

or, x²+ 7x=x²+ 2x + 1

or, 5x=1

∴ x = $\frac{1}{5}$