Power of a Quotient Rule for Exponents - Examples, Exercises and Solutions

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.

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.

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.

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

$\frac{c^m}{c^n}=c^{m-n}$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 mentioned power property:

$\frac{a^a}{a^b}=a^{a-b}$Therefore, the correct answer is option D.

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.

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.

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.

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).

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.

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.

Note that we need to perform division between two terms with identical bases, therefore we will use the law of exponents for division between terms with identical bases:

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

Let's return to the problem and apply the above law of exponents to each term in the sum separately:

$\frac{a^x}{a^y}+\frac{a^2}{a^x}=a^{x-y}+a^{2-x}$Therefore the correct answer is A.

Here we have division between two terms with identical bases, therefore we will use the power property to divide 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.

Let's go back to the problem and apply the power property to each term of the exercise separately:

$\frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=b^{y-x}-b^{z-3}$Therefore, the correct answer is option A.

Due to the fact that the numerator and the denominator of the fraction have terms with identical bases, we will begin by applying the law of exponents for the division of terms with identical bases:

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

$\frac{3a^2}{2a}=\frac{3}{2}\cdot a^{2-1}=\frac{3}{2}\cdot a^1$In the first step we simplify the numerical part of the fraction. This is a simple and intuitive step which makes it easier to work with the said fraction.

$\frac{3a^2}{2a}=\frac{3}{2}\cdot\frac{a^2}{a}=\frac{3}{2}\cdot a^{2-1}=\ldots$Let's return to the problem, remember that any number raised to 1 is equal to the number itself, that is:

$b^1=b$Thus we apply it to the problem:

$\frac{3}{2}\cdot a^1=\frac{3}{2}\cdot a=1\frac{1}{2}a$In the last step we convert the fraction into a mixed fraction.

$$Therefore, the correct answer is option D.

Due to the fact that the numerator and the denominator of the fraction have terms with identical bases, we will begin by applying the law of exponents for the division of terms with identical bases:

$\frac{b^m}{b^n}=b^{m-n}$We begin by applying the formula to the given problem:

$\frac{4a^5}{2a^3}=2\cdot a^{5-3}=2\cdot a^2$In the first step we simplify the numerical part of the fraction. This is simple to do and makes it easier to work with the said fraction.

$\frac{4a^5}{2a^3}=\frac{4}{2}\cdot\frac{a^5}{a^3}=2\cdot a^{5-3}=\ldots$We obtain the following answer:

$2a^2$

$$Therefore, the correct answer is option A.

First we rewrite the first expression on the left of the problem as a fraction:

$\frac{a^2}{a}+a^3\cdot a^5$Then we use two properties of exponentiation, to multiply and divide terms with identical bases:

1.

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

2.

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

Returning to the problem and applying the two properties of exponentiation mentioned earlier:

$\frac{a^2}{a}+a^3\cdot a^5=a^{2-1}+a^{3+5}=a^1+a^8=a+a^8$

Later on, keep in mind that we need to factor the expression we obtained in the last step by extracting the common factor,

Therefore, we extract from outside the parentheses the greatest common divisor to the two terms which are:

$a$ We obtain the expression:

$a+a^8=a(1+a^7)$ when we use the property of exponentiation mentioned earlier in A.

$$$a^8=a^{1+7}=a^1\cdot a^7=a\cdot a^7$

Summarizing the solution to the problem and all the steps, we obtained the following:

$\frac{a^2}{a}+a^3\cdot a^5=a(1+a^{7)}$

Therefore, the correct answer is option b.