Power of a Quotient Rule for Exponents: Variables in the exponent of the power

Let's first deal with the first term in the multiplication, noting that the terms in the numerator and denominator have identical bases, so we'll use the power rule for division between terms with the same base:

$\frac{a^m}{a^n}=a^{m-n}$We'll apply for the first term in the expression:

$\frac{a^{3b}}{a^{2b}}\cdot a^b=a^{3b-2b}\cdot a^b=a^b\cdot a^b$where we also simplified the expression we got as a result of subtracting the exponents of the first term,

Next, we'll notice that the two terms in the multiplication have identical bases, so we'll use the power rule for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We'll apply to the problem:

$a^b\cdot a^b=a^{b+b}=a^{2b}$Therefore, the correct answer is A.

Note that in this problem there is a fraction where both the numerator and denominator contain terms with identical bases, therefore we will use the division law between terms with identical bases 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 directly used the above exponents law and performed a simple subtraction between the exponent of the term in the numerator and the exponent of the term in the denominator, while 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.

We'll continue by simplifying the exponential expression, and use the distribution law to expand the parentheses, while remembering that the (-) sign before the parentheses is actually multiplication by minus 1:

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

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

We obtained the most simplified expression, so we are done.

Therefore, the correct answer is B.

Keep in mind that in the question there is a fraction containing identical terms in its numerator and denominator. Therefore, we can use the distributive property of division to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$
We apply this to our problem and simplify the first term:

$\frac{a^{xy}}{a^{3xy}}-a^{2xy}=a^{xy-3xy}-a^{2xy}=a^{-2xy}-a^{2xy}$

In the second step, calculate the result of the subtraction operation in the exponent to obtain:

$a^{-2xy}-a^{2xy}$

Therefore, the correct answer is D.

The problem involves multiplication between two fractions, so first we'll apply the rule for multiplying fractions which states that multiplication between two fractions is calculated by putting one fraction over a line by multiplying the numerators together and multiplying the denominators together, mathematically:

$\frac{x}{y}\cdot\frac{w}{m}=\frac{x\cdot w}{y\cdot m}$

Let's apply it to the problem:

$\frac{a^{2x}}{a^{y}}\cdot\frac{a^{2y}}{a^{-y}}=\frac{a^{2x}\cdot a^{2y}}{a^y\cdot a^{-y}}$

Next, we'll notice that both in the numerator separately and in the denominator separately there is multiplication between terms with identical bases, so we'll use the power law for multiplying terms with identical bases:

$a^m\cdot a^n=a^{m+n}$Let's apply it separately to the numerator and denominator in the problem:

$\frac{a^{2x}\cdot a^{2y}}{a^y\cdot a^{-y}}=\frac{a^{2x+2y}}{a^{y+(-y)}}=\frac{a^{2x+2y}}{a^0}$

Next, we'll remember that any number to the power of 0 equals 1, mathematically:

$a^0=1$

So let's return to the problem:

$\frac{a^{2x+2y}}{a^{0}}=\frac{a^{2x+2y}}{1}=a^{2x+2y}$

Where we actually used the fact that division by 1 doesn't change the value of the number, meaning mathematically:

$X:1=\frac{X}{1}=X$

Now let's examine the result we got above:

$a^{2x+2y}\text{ }$

In terms of simplification using the laws of exponents we have indeed finished since this is the most simplified expression,

but it's worth noting that in the exponent we got an expression that can be factored using common factor extraction:

$2x+2y=2(x+y)$

In this case the common factor is the number 2,

Let's return to the result of the expression simplification, we got:

$a^{2x+2y}=a^{2(x+y)}$Therefore the correct answer is C.