Power quotient - Examples, Exercises and Solutions

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

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$$$

Simplify the following:

$\frac{a^a}{a^b}=$

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.

$a^{a-b}$

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.

$a^2$

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 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).

$a^{9-x}$

Solve the following exercise

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

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

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

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

$a^{7y-5x}$

Solve the following:

$\frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=$

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.

$b^{y-x}-b^{z-3}$

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.

$a^{10}$

Solve the following exercise:

$\frac{14a^{-3}}{7a^{-3}}=$

Let's consider that the numerator and the denominator of the fraction have terms with identical bases, therefore we will use the division between terms with identical bases:

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

$\frac{14a^{-3}}{7a^{-3}}=2a^{-3-(-3)}=2a^{-3+3}=2a^0$When in the first step we reduce the numerical part of the fraction, this operation is correct and intuitive because it is always possible to previously note the mentioned fraction as a product of fractions and simplify:

$\frac{14a^{-3}}{7a^{-3}}=\frac{14}{7}\cdot\frac{a^{-3}}{a^{-3}}=2a^{-3-(-3)}=\ldots$We return to the problem and remember that any number raised to the 0th power is 1, that is:

$b^0=1$Therefore, in the problem we obtain:

$2a^0=2\cdot1=2$Therefore, the correct answer is option B.

$2$

Solve the following exercise:

$\frac{-3a^{-2}}{-6a^{-6}}=$

Let's consider that the numerator and the denominator of the fraction have terms with identical bases, therefore we will use the property of division between terms with identical bases:

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

$\frac{-3a^{-2}}{-6a^{-6}}=\frac{1}{2}\cdot a^{-2-(-6)}=\frac{1}{2}\cdot a^{-2+6}=\frac{1}{2}\cdot a^4$When in the first step we reduce the numerical part of the fraction, this operation is correct and intuitive because it is always possible to previously write the mentioned fraction as a product of fractions and reduce:

$\frac{-3a^{-2}}{-6a^{-6}}=\frac{-3}{-6}\cdot\frac{a^{-2}}{a^{-6}}=\frac{1}{2}\cdot\frac{a^{-2}}{a^{-6}}=\ldots$We return to the problem, we obtain the expression:

$\frac{1}{2}\cdot a^4$Therefore, the correct answer is option C.

$\frac{1}{2}a^4$

Solve the exercise:

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

Let's consider that the numerator and the denominator of the fraction have terms with identical bases, therefore we will use the property of division between terms with identical bases:

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

$\frac{3a^2}{2a}=\frac{3}{2}\cdot a^{2-1}=\frac{3}{2}\cdot a^1$When in the first step we reduce the numerical part of the fraction, this operation is correct and intuitive because it is always possible to previously note the mentioned fraction as a product of fractions and reduce:

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

$b^1=b$We apply it to the problem:

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

$$Therefore, the correct answer is option D.

$1 \frac{1}{2}a$

Solve the exercise:

$\frac{4a^5}{2a^3}=$

Let's note that the numerator and the denominator of the fraction have terms with identical bases, therefore we will use the property of division between terms with identical bases:

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

$\frac{4a^5}{2a^3}=2\cdot a^{5-3}=2\cdot a^2$When in the first step we simplify the numerical part of the fraction, this operation is correct and intuitive because it is always possible to previously note the mentioned fraction as a product of fractions and reduce:

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

$2a^2$

$$Therefore, the correct answer is option A.

$2a^2$

Complete the exercise:

$\frac{12b^4}{4b^{-5}}=$

Let's consider that the numerator and the denominator of the fraction have terms with identical bases, therefore we will use the property of division between terms with identical bases:

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

$\frac{12b^4}{4b^{-5}}=3\cdot b^{4-(-5)}=3\cdot b^{4+5}=3b^9$When in the first step we simplify the numerical part of the fraction, this operation is correct and intuitive because it is always possible to previously note the mentioned fraction as a product of fractions and reduce:

$\frac{12b^4}{4b^{-5}}=\frac{12}{4}\cdot\frac{b^4}{b^{-5}}=3\cdot b^{4-(-5)}=\ldots$We return to the problem. We obtained that the simplified expression is:

$3b^9$

$$Therefore, the correct answer is option D.

$3b^9$