INDICES

Indices

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

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} x^{2 – 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{\: }}$

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

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

= a^{3 – 1} * b ^{-1 + 1} * c^{2 – 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.

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