# INDICES

Indices

Laws of the indices

1. am× an=am+n

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

3. a0= 1

4. (am)n = amn

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

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

8. If am= bnthen,a= bn/m

9. if am= bnthen, m=n

10. am= bnthen, a =b

Exponential equation

The equation in which variables appears as the components of base is called exponential equation.

Examples

1.

(5x-1y2)-2÷ (25x2y-1)2

= $\frac{1}{{25}}$.x2.y-4÷54x4y-2

= 5-2x2y-4÷ 54x4y-2

= 5-2 – 4 x2 – 4 y-4 + 2

= 5-6x-2y-3

= $\frac{1}{{{5^6}{{\rm{x}}^2}{{\rm{y}}^2}}}$

2.

$\sqrt {{{\rm{a}}^6}{{\rm{b}}^{ - 2}}{{\rm{c}}^4}}$÷$\sqrt[4]{{({{\rm{a}}^4}{{\rm{b}}^{ - 4}}{{\rm{c}}^8}}}{\rm{\: }}$

= ${{\rm{a}}^{\frac{6}{2}}}{\rm{*}}{{\rm{b}}^{ - \frac{2}{2}}}{\rm{*}}{{\rm{c}}^{\frac{4}{2}}}$÷$\left( {{{\rm{a}}^{\frac{4}{4}}}{\rm{*}}{{\rm{b}}^{ - \frac{4}{4}}}{\rm{*}}{{\rm{c}}^{\frac{8}{4}}}} \right)$

= ${{\rm{a}}^3}{\rm{*}}{{\rm{b}}^{ - 1}}{\rm{*}}{{\rm{c}}^2}$÷$\left( {{{\rm{a}}^1}{\rm{*}}{{\rm{b}}^{ - 1}}{\rm{*}}{{\rm{c}}^2}} \right)$

= a3 – 1 * b -1 + 1 * c2 – 2.

= a2.

3.

$\frac{{{3^{3{\rm{a}} + 2}} - {3^{3{\rm{a}} + 1}}}}{{6{\rm{*}}{{27}^{\rm{a}}}}}$

= $\frac{{{3^{3{\rm{a}} + 1}}.3 - {3^{3{\rm{a}} + 1}}}}{{3{\rm{*}}2{\rm{*}}{3^{3{\rm{a}}}}}}$

= $\frac{{{3^{3{\rm{a}} + 1}}\left( {3 - 1} \right)}}{{{3^{3{\rm{a}} + 1}}{\rm{*}}2}}$

=$\frac{{3 - 1}}{2}$

= 1.

4.

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

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

5.

4x-6.2x+1+32 = 0

Soln

4x-6.2x+1+32 = 0

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

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

let , 2x=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

2x = 22,

x = 2

2x =8

x= 3

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