8th Grade - Laws of Exponents

Introduction

Have you seen any number with another number (positive or negative) on the top right? You might have already; those are called Exponents.
$2^{3} or 2^{- 4}$ are some examples of exponents, also called exponent expressions.
Exponents are nothing but repeated multiplication.
In $2^{3}$ , 2 is the base, and 3 is the exponent. In the case of $3 {(x)}^{2}$ , 3 is called the coefficient, x is the base, and 2 is the exponent.
$2^{3}$ is read as two to the third power and also as two cubed.
$7^{4}$ is read as seven to the fourth power.
Exponents can be positive or negative.

$3^{- 4}$ is an example of negative exponents.
Negative exponents can be solved by finding a reciprocal (dropping that number under 1)
For example, $3^{- 4}$ is same as $\frac{1}{3^{4}}$ which is same as $\frac{1}{81}$

If the base and exponents are the same, you only need to add/subtract the coefficient.
$2 x^{2} + 3 x^{2}$ can be added, and the result is $5 x^{2}$ .
$4 x^{2} - 2 x^{2}$ can be subtracted, and the result is $2 x^{2}$ .
$3 x^{2} + 4 x^{4}$ can't be added as the exponents are different.
$3 x^{2} - 2 y^{2}$ can't be subtracted as the bases are different.
$2 {(3)}^{3} + 3 {(2)}^{2}$ can be added only by evaluating $2 {(3)}^{3}$ and $3 {(2)}^{2}$ separately and then adding. So it is 18 + 12, which is 30.

To multiply exponents with the same base, keep the base and add the exponent.
$a^{x} \times a^{y}$ is the same as $a^{(x + y)}$ .
$3^{2} \times 3^{4}$ is the same as $3^{(2 + 4)}$ which equates to $3^{6}$ .
If the bases are different, simplify each expression and then perform the multiplication.

To divide exponents with the same base, keep the base and subtract the exponent.
$a^{x} \div a^{y}$ is the same as $a^{(x - y)}$ .
$3^{4} \div 3^{2}$ is the same as $3^{(4 - 2)}$ which equates to $3^{2}$ .
If the bases are different, simplify each expression and then perform the division.