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

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 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.

Using the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Here, we have $\frac{3^5}{3^2} = 3^{5-2}$

Simplifying, we get $3^3$

Using the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Here, we have $\frac{5^6}{5^4} = 5^{6-4}$. Simplifying, we get $5^2$.

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, for good order, let's write the expression as a fraction:

$\frac{a^5}{a^4}$ Then we'll recall the law of exponents for division between terms with equal bases:

$\frac{c^m}{c^n}=c^{m-n}$ and we'll apply this law to our problem:

$\frac{a^5}{a^4}=a^{5-4}=a^1=a$where in the second step we calculated the result of the subtraction in the exponent and then used the fact that any number raised to the power of 1 equals the number itself, meaning that:

$X^1=X$ We got that: $\frac{a^5}{a^4}=a$therefore the correct answer is a.

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.

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 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.

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.

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.

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.

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.