Simplify the Expression: a^(7+x)/a^(10-2x) Using Exponent Rules

In this problem there is a fraction where both the numerator and denominator contain terms with identical bases. Therefore we will apply the division law between terms with identical bases in order to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$

Let's apply the aforementioned law:

$\frac{a^{7+x}}{a^{10-2x}}=a^{7+x-(10-2x)}$

In the first stage, we used the above exponents law and then proceeded to perform a simple subtraction between the exponent of the term in the numerator and the exponent of the term in the denominator, whilst using a common base. Since the term's exponent in the denominator is a two-term algebraic expression, we performed the subtraction carefully using parentheses.

Proceed to simplify the exponential expression, and apply the distribution law in order to expand the parentheses, whilst remembering that the (-) sign before the parentheses is actually a multiplication operation by minus 1:

$a^{7+x-(10-2x)}=a^{7+x-10+2x}=a^{-3+3x}$

In the second stage we combined like terms in the exponent.

In doing so we obtained the most simplified form of our expression.

Therefore, the correct answer is B.