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**Hint:**Here we will simply use the main equation to get the value of \[8 \circ 3\]. First, we will compare \[8 \circ 3\] is similar to \[a \circ b\] and find the values of the variables. Then we will substitute the values of the variables in the given equation. We will simplify the equation using BODMAS to get the value of \[8 \circ 3\].

**Complete step-by-step answer:**Given equation is \[a \circ b = {a^2} + ba - 16b \div a\].

Here, \[8 \circ 3\] is similar to \[a \circ b\]. So we will compare them to find the value of \[a\] and \[b\].

So, by comparing it to the LHS of the given equation we can say that \[a = 8\] and \[b = 3\].

Now substituting \[a = 8\] and \[b = 3\] in \[a \circ b = {a^2} + ba - 16b \div a\], we get

\[8 \circ 3 = {8^2} + \left( {3 \times 8} \right) - \left( {16 \times 3 \div 8} \right)\]

\[8 \circ 3 = {8^2} + \left( {3 \times 8} \right) - \left( {16 \times \dfrac{3}{8}} \right)\]

Now we will solve this above equation to get the value of \[8 \circ 3\].

Multiplying the terms in the bracket, we get

\[ \Rightarrow 8 \circ 3 = 64 + 24 - \left( {\dfrac{{48}}{8}} \right)\]

Dividing 48 by 8, we get

\[ \Rightarrow 8 \circ 3 = 64 + 24 - 6\]

Adding and subtracting the like terms, we get

\[ \Rightarrow 8 \circ 3 = 82\]

Hence the value of \[8 \circ 3\] is equal to 82.

**So, option E is the correct option.**

**Note:**Here in this question, where a function equation is given we have to simply substitute the values in it to get the value of the desired equation. This function equation is defined in such a way that it applies to all the values of the variable.

Here we have used the BODMAS rule to solve the equation. BODMAS stands for B-Brackets, O-Of, D-Division, M-Multiplication, A-Addition, S-Subtraction. This means we will first solve the brackets, then division, then multiplication, then addition and in the last subtraction.

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