Power of a Quotient Rule for Exponents: Applying the formula

$\frac{2^4}{2^3}=$

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$We apply it in the problem:

$\frac{2^4}{2^3}=2^{4-3}=2^1$Remember that any number raised to the 1st power is equal to the number itself, meaning that:

$b^1=b$Therefore, in the problem we obtain:

$2^1=2$Therefore, the correct answer is option a.

$2$

$\frac{81}{3^2}=$

First, we recognize that 81 is a power of the number 3, which means that:

$3^4=81$We replace in the problem:

$\frac{81}{3^2}=\frac{3^4}{3^2}$Keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$We apply it in the problem:

$\frac{3^4}{3^2}=3^{4-2}=3^2$$$Therefore, the correct answer is option b.

$3^2$$$

$\frac{9^9}{9^3}=$

Note that in the fraction and its denominator, there are terms with the same base, so we will use the law of exponents for division between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$ Let's apply it to the problem:

$\frac{9^9}{9^3}=9^{9-3}=9^6$$$Therefore, the correct answer is b.

$9^6$

Simplify the following:

$\frac{a^7}{a^3}=$

Sincw a division operation between two terms with identical bases is required, we will use the power property to divide between terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$Note that using this property is only possible when the division is performed between terms with identical bases.

We return to the problem and apply the power property:

$\frac{a^7}{a^3}=a^{7-3}=a^4$Therefore, the correct answer is option C.

$a^4$

Simplify the following:

$\frac{a^3}{a^1}=$

Since a division operation between two terms with identical bases is required, we will use the power property to divide between terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$Note that using this property is only possible when the division is performed between terms with identical bases.

We return to the problem and apply the power property:

$\frac{a^3}{a^1}=a^{3-1}=a^2$Therefore, the correct answer is option A.

$a^2$

Simplify the following:

$\frac{a^5}{a^3}=$

Since a division operation between two terms with identical bases is required, we will use the power property to divide between terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$Note that using this property is only possible when the division is carried out between terms with identical bases.

We return to the problem and apply the mentioned power property:

$\frac{a^5}{a^3}=a^{5-3}=a^2$Therefore, the correct answer is option A.

$a^2$

Solve the following exercise

$\frac{a^{7y}}{a^{5x}}$

Let's consider that in the given problem there is a fraction in both the numerator and denominator with terms of identical bases. Hence we use the property of division between terms of identical bases in order to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$We apply the previously mentioned property to the problem:

$\frac{a^{7y}}{a^{5x}}=a^{7y-5x}$Therefore, the correct answer is option A.

$a^{7y-5x}$

Simplify the following expression:

$\frac{a^9}{a^x}$

In the question there is a fraction that has terms with identical bases in its numerator and denominator. Therefore, so we can use the distributive property of division to solve the exercise:

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

We apply the previously distributive property to the problem:

$\frac{a^9}{a^x}=a^{9-x}$

Therefore, the correct answer is (c).

$a^{9-x}$

Simplify the following:

$\frac{a^4}{a^{-6}}=$

Since a division operation between two terms with identical bases is required, we will use the power property to divide terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$$\frac{c^m}{c^n}=c^{m-n}$Note that using this property is only possible when the division is performed between terms with identical bases.

We return to the problem and apply the power property:

$\frac{a^4}{a^{-6}}=a^{4-(-6)}=a^{4+6}=a^{10}$Therefore, the correct answer is option C.

$a^{10}$

$\frac{1}{\frac{X^7}{X^6}}=$

First, we will focus on the exercise with a fraction in the denominator. We will solve it using the formula:

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

$\frac{x^7}{x^6}=x^{7-6}=x^1$

Therefore, we get:

$\frac{1}{x}$

We know that a product raised to the 0 is equal to 1 and therefore:

$\frac{x^0}{x^1}=x^{(0-1)}=x^{-1}$

$X^{-1}$

Choose the expression that is equal to the following:

$a^5:a^4$

$a$