Solve Complex Exponent Expression: (a^12/a^20 × a^7)/(a^4/a^7)

First, let’s tidy up the problem and rewrite it in a more familiar form. Recall that division by a fraction is the same as multiplication by its reciprocal. To find the reciprocal of a fraction, we simply swap its numerator and denominator. Mathematically, instead of writing:

$:\frac{x}{y}$we can always write:

$\cdot\frac{y}{x}$Let's apply this to the problem:

$\frac{a^{12}}{a^{20}}\cdot a^7:\frac{a^4}{a^7}=\frac{a^{12}}{a^{20}}\cdot a^7\cdot\frac{a^7}{a^4}$

From here the solution becomes clear, let's continue with the multiplication of fractions:

$\frac{a^{12}}{a^{20}}\cdot a^7\cdot\frac{a^7}{a^4}=\frac{a^{12}\cdot a^7\cdot a^7}{a^{20}\cdot a^4}$

To multiply the fraction, we actually used the fact that any number can always be written as a fraction by writing it with denominator 1, for example:

$X=\frac{X}{1}$So we actually performed:

$\frac{a^{12}}{a^{20}}\cdot a^7\cdot\frac{a^7}{a^4}=\frac{a^{12}}{a^{20}}\cdot\frac{a^7}{1}\cdot\frac{a^7}{a^4}=\frac{a^{12}\cdot a^7\cdot a^7}{a^{20}\cdot1\cdot a^4}=\frac{a^{12}\cdot a^7\cdot a^7}{a^{20}\cdot a^4}$

Therefore, multiplication by a fraction is simply multiplying the numerator.

At this point, we’ve finished the cosmetic rearrangement and can return to the laws of exponents. Notice that in both the numerator and the denominator of the fraction, all terms share the same base. Therefore, we can apply the law of exponents for multiplication with identical bases:

$a^m\cdot a^n=a^{m+n}$

It’s important to note that this law applies to any number of factors in a product, not just two. For example, when multiplying three terms with the same base, we have:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$

When we used the above law of exponents twice, we can also perform the same calculation for four terms in multiplication, five, and so on...

Let's return to the problem, let's apply the expression we got to the numerator and denominator:

$\frac{a^{12}\cdot a^7\cdot a^7}{a^{20}\cdot a^4}=\frac{a^{12+7+7}}{a^{20+4}}=\frac{a^{26}}{a^{24}}$

From here we'll use the law of division between terms with identical bases:

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

Let's apply to the problem:

$\frac{a^{26}}{a^{24}}=a^{26-24}=a^2$

Therefore the correct answer is A.

Note:

It is highly recommended when encountering a messy expression like in the problem to do some "cosmetic work" using the multiplication by reciprocal law mentioned above - as we did at the beginning of the solution, in order to get a familiar form for which there will be no doubts about using the laws or the order of operations. In general - always try to reach an expression with only one "level" of fraction lines, for example:

$\frac{x}{y}:\frac{z}{w}=\frac{\frac{x}{y}}{\frac{z}{w}}=\frac{x}{y}\cdot\frac{w}{z}=\frac{x\cdot w}{y\cdot z}$

This way the algebraic expressions are clearer and their simplification is easier.