Simplify the following mathematical expression:

$\frac{27{\left(1.33\right)}^3-1^3}{{\left(3.99\right)}^2+4.99}$ = ?

The correct option is 2.99.

To simplify the given mathematical expression efficiently without performing tedious decimal calculations, we can use an algebraic identity.
The given expression is:

$\frac{27{\left(1.33\right)}^3-1^3}{{\left(3.99\right)}^2+4.99}$

Step 1: Rewrite the numerator
Observe that $27=3^3$.
Using the exponent power rule $x^n·y^n={(x·y)}^n$, we can rewrite the term $27{\left(1.33\right)}^3$ as follows:

$27{\left(1.33\right)}^3=3^3\times {\left(1.33\right)}^3={(3\times 1.33)}^3={\left(3.99\right)}^3$

Therefore, the numerator becomes ${\left(3.99\right)}^3-1^3$.

Step 2: Rewrite the denominator
Now, examine the denominator ${\left(3.99\right)}^2+4.99$.
Notice that $4.99=3.99+1=(3.99\times 1)+1^2$.
So the denominator can be expressed as:

${\left(3.99\right)}^2+4.99={\left(3.99\right)}^2+(3.99\times 1)+1^2$

Step 3: Apply the difference of cubes identity
Recall the algebraic identity for the difference of two cubes:
$a^3-b^3=(a-b)(a^2+ab+b^2)$
By setting $a=3.99$ and $b=1$, the numerator expands to:

${\left(3.99\right)}^3-1^3=(3.99-1)[{\left(3.99\right)}^2+(3.99\times 1)+1^2]$

Step 4: Substitute and simplify the fraction
Substitute both the expanded numerator and rewritten denominator back into the original fraction:

$\frac{27{\left(1.33\right)}^3-1^3}{{\left(3.99\right)}^2+4.99}=\frac{(3.99-1)[{\left(3.99\right)}^2+3.99+1]}{{\left(3.99\right)}^2+3.99+1}$

The expression ${\left(3.99\right)}^2+3.99+1$ in the numerator and denominator cancels out, leaving:

$=3.99-1=2.99$

Thus, the final simplified answer is 2.99.