How do you simplify \[\dfrac{{9{b^6} - 6{b^3} + 2}}{{3{b^3}}}\]

Hint: The equation is an algebraic equation, where the algebraic equation is the combination of constants and variables. Here in this question we have to simplify the given algebraic equation, the algebraic equation is in the form of fraction and by using the law of exponents we can solve the given question.

Complete step by step solution:
The algebraic expression is an expression which consists of variables and constants with the arithmetic operations. Since it contains the exponents form we can use the law of exponents for the given algebraic expression ,we apply simple methods. Since by solving these types of expression we get the simplified form.
Now consider the given algebraic expression
 \[\dfrac{{9{b^6} - 6{b^3} + 2}}{{3{b^3}}}\]
The denominator is implemented to the all terms which are present in the numerator. So the given algebraic expression is written as
 \[ \Rightarrow \dfrac{{9{b^6}}}{{3{b^3}}} - \dfrac{{6{b^3}}}{{3{b^3}}} + \dfrac{2}{{3{b^3}}}\]
On simplifying we get
  \[ \Rightarrow \dfrac{{3{b^6}}}{{{b^3}}} - \dfrac{{2{b^3}}}{{{b^3}}} + \dfrac{2}{{3{b^3}}}\]
By the law of exponents we have \[{a^{m - n}} = \dfrac{{{a^m}}}{{{a^n}}}\] , using this law of exponent the algebraic expression is written as
 \[ \Rightarrow 3{b^{6 - 3}} - 2{b^{3 - 3}} + \dfrac{2}{{3{b^3}}}\]
On simplifying the some terms we have
 \[ \Rightarrow 3{b^3} - 2{b^0} + \dfrac{2}{{3{b^3}}}\]
Any number or any variable which has a power 0 then its value is 1. Or by the law of exponent \[{a^0} = 1\] , now the algebraic expression is written as
 \[ \Rightarrow 3{b^3} - 2 + \dfrac{2}{{3{b^3}}}\]
In the term let we take the exponent number which is present in the denominator to the numerator and therefore the algebraic expression is written as
 \[ \Rightarrow 3{b^3} - 2 + \dfrac{{2{b^{ - 3}}}}{3}\]
Hence the algebraic expression is simplified.
So, the correct answer is “ \[ 3{b^3} - 2 + \dfrac{{2{b^{ - 3}}}}{3}\] .

Note: The algebraic equation or an expression is a combination of variables and constants, it also contains the coefficient. The alphabets are known as variables. The x, y, z etc., are called as variables. The numerals are known as constants. The numeral of a variable is known as co-efficient. We have 3 types of algebraic expressions namely monomial expression, binomial expression and polynomial expression. By using the law of exponent and simple arithmetic operations, we can solve the equation.