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Exponents with Variables

Exponent of a variable is the power raised to it. Exponent implies the repeated multiplication to itself. The number of times a variable is multiplied, is raised as its exponent or power. We can define it by the following relation:

Exponent of Variable
Where, x is a variable and n is its exponent.
$x^{n}$ is read as "x raised to the exponent of n", "x raised to the power of n" or more casually "x to the power n". Exponent of 2 and 3 over a variable are pronounced differently, such as $x^{2}$ and $x^{3}$ are read as "x squared" and "x cubed" respectively.

There are few important laws that are to be followed while dealing with exponents. These laws are as follows (Let us consider two variables x and y and exponents m and n):

1. $x^{0}=1$

2. $x^{-m}$=$\frac{1}{x^{m}}$

3. $x^{m}.x^{n}=x^{m+n}$

4. $\frac{x^{m}}{x^{n}}$=$x^{m-n}$

5. $(x^{m})^{n}=x^{mn}$

6. $x^{m}y^{m}=(xy)^{m}$

7. $\frac{x^{m}}{y^{m}}$=($\frac{x}{y}$)$^{m}$

8. $x^{\frac{m}{n}}$=$\sqrt[n]{x^{m}}$

An example based on exponents of variables and their laws is given below:

Example: Solve $(-2x^{-1}y^{3})^{2}.(-5x^{2}y^{-2}z)$.

Solution: $(-2x^{-1}y^{3})^{2}.(-5x^{2}y^{-2}z)$

= $(-2)^{2}.(x^{-1})^{2}.(y^{3})^{2}.(-5).x^{2}.y^{-2}.z$

= $4.x^{-2}.y^{6}.(-5).x^{2}.y^{-2}.z$

= $-20.\frac{1}{x^{2}}.y^{6}.x^{2}.\frac{1}{y^{2}}.z$

= $-20.x^{2-2}.y^{6-2}.z$

= $-20x^{0}y^{4}z$

= $-20y^{4}z$

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