Simplify (a⁷/a⁵) × (a⁻²/a⁻⁴) + (a⁷)³: Complete Exponent Expression

a. We'll start by handling the first expression from the left, which is a multiplication of fractions. However, we won't perform the fraction multiplication; instead, we'll notice that in each of the fractions in the multiplication, there are terms in both numerator and denominator with identical bases,

therefore we'll use the power law for division between terms with identical bases:

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

We'll apply this to our problem in the first expression from the left and apply the above power law separately for each term in the fraction multiplication:

$\frac{a^7}{a^5}\cdot\frac{a^{-2}}{a^{-4}}=a^{7-5}\cdot a^{-2-(-4)}=a^2\cdot a^{-2+4}=a^2\cdot a^2$

Next, we'll notice that we got an algebraic expression that is actually the term multiplied by itself, therefore from the definition of power we can write the expression as a power:

$a^2\cdot a^2=(a^2)^2$

(Of course we could also apply the multiplication law between terms with identical bases to the above expression, but here we want to also recall the power of a power law, so we'll choose this way),

Now let's recall the power of a power law:

$(c^m)^n=c^{m\cdot n}$

and apply it to the expression we got:

$(a^2)^2=a^{2\cdot2}=a^4$

We've finished handling the first expression from the left, meaning we got that:

$\frac{a^7}{a^5}\cdot\frac{a^{-2}}{a^{-4}}=a^4$

We'll remember this result and move on to the next expression.

b. For the second expression from the left, we'll apply again the power of a power law mentioned in section a:

$(a^7)^3=a^{7\cdot3}=a^{21}$

and we've finished handling this expression as well.

Let's summarize the two solution sections a and b into the simplified expression result:

$\frac{a^7}{a^5}\cdot\frac{a^{-2}}{a^{-4}}+(a^7)^3=a^4+a^{21}$

Therefore the correct answer is b.