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Solving Equations with Exponents

Exponent is referred as power of a variable. The expression $x^{m}$ is an exponential expression, where "x" is known as "base" and "m" is called "exponent".
$x^{m}=x*x*x*....*x(m\ times)$An equation containing exponential expression is known as exponential equation.

For example: $x^{\frac{4}{3}}$=$256$.
There are few rules of exponents which are to be followed while solving exponential equations. These rules are known as laws of exponents which are as follows:
(1) If bases are equal, then exponents will also be equal.

Exponents
(2) While multiplying two exponential expressions with same base, exponents will be added on that base.
Multiplication of Exponents
(3) While dividing two exponential expressions with same base, the exponent of denominator will be subtracted from the exponent of numerator on that base.
Division of Exponents
(4) While multiplying two exponential expressions with same exponent on different bases, the multiplication of bases will be raised to that exponent.
Multiplication of Equal Exponents
(5) While dividing two exponential expression with same exponent on different bases, the fraction will be raised to that exponent.
Division of Equal Exponents
(6) When an exponential expression is raised by an exponent, both the exponents are multiplied.
Exponent Rule
(7) The negative exponent of a variable follows the following law:
Negative Exponent
(8) Fractional exponent of a variable follows the following rule:
Fractional Exponent
Let us understand laws of exponents with few examples:

Example 1: Solve $2^{3x+1}=8^{2x}$ for x.
Solution: $2^{3x+1}=8^{2x}$
$2^{3x+1}=(2^{3})^{2x}$
$2^{3x+1}=2^{6x}$  (Using law $(a^m)^n = (a)^{mn}$)
$3x+1=6x$ (Using law, if $a^m = a^n$ then m = n)
$3x=1$
$x=$$\frac{1}{3}$

Example 2: Solve $3^{2a}-1=26$.
Solution: $3^{2a}-1=26$.
$3^{2a}=26+1$
$3^{2a}=27$
$3^{2a}=3^{3}$
$2a=3$ (Using law, if $a^m = a^n$ then m = n)
$a=$$\frac{3}{2}$

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