Power quotient - Examples, Exercises and Solutions

Division of Exponents with the Same Base

When we encounter exercises or expressions with terms that have the same base and between them the sign of division or fraction line, we can subtract the exponents.
We will subtract the exponent in the denominator from the exponent in the numerator.
That is:
"exponent of the denominator - exponent of the numerator" = new exponent
The result obtained from the subtraction is the new exponent and we will apply it to the original base.

Formula of the property:

aman=a(mn)\frac {a^m}{a^n} =a^{(m-n)}

This property also concerns algebraic expressions.

Suggested Topics to Practice in Advance

  1. Power of a Power
  2. Exponent of a Multiplication
  3. Power of a Quotient
  4. Multiplying Exponents with the Same Base

Practice Power quotient

Exercise #1

8132= \frac{81}{3^2}=

Video Solution

Step-by-Step Solution

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

34=81 3^4=81 We replace in the problem:

8132=3432 \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:

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

3432=342=32 \frac{3^4}{3^2}=3^{4-2}=3^2 Therefore, the correct answer is option b.

Answer

32 3^2

Exercise #2

2423= \frac{2^4}{2^3}=

Video Solution

Step-by-Step Solution

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:

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

2423=243=21 \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:

b1=b b^1=b Therefore, in the problem we obtain:

21=2 2^1=2 Therefore, the correct answer is option a.

Answer

2 2

Exercise #3

Simplify the following expression:

a9ax \frac{a^9}{a^x}

Video Solution

Step-by-Step Solution

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:

cmcn=cmn \frac{c^m}{c^n}=c^{m-n}

We apply the previously distributive property to the problem:

a9ax=a9x \frac{a^9}{a^x}=a^{9-x}

Therefore, the correct answer is (c).

Answer

a9x a^{9-x}

Exercise #4

Solve the following exercise

a7ya5x \frac{a^{7y}}{a^{5x}}

Video Solution

Step-by-Step Solution

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:

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

a7ya5x=a7y5x \frac{a^{7y}}{a^{5x}}=a^{7y-5x} Therefore, the correct answer is option A.

Answer

a7y5x a^{7y-5x}

Exercise #5

Simplify the following:

a5a3= \frac{a^5}{a^3}=

Video Solution

Step-by-Step Solution

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:

cmcn=cmn \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:

a5a3=a53=a2 \frac{a^5}{a^3}=a^{5-3}=a^2 Therefore, the correct answer is option A.

Answer

a2 a^2

Exercise #1

Simplify the following:


a3a1= \frac{a^3}{a^1}=

Video Solution

Step-by-Step Solution

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:

cmcn=cmn \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:

a3a1=a31=a2 \frac{a^3}{a^1}=a^{3-1}=a^2 Therefore, the correct answer is option A.

Answer

a2 a^2

Exercise #2

Simplify the following:

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

Video Solution

Step-by-Step Solution

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

cmcn=cmn \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:

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

Answer

aab a^{a-b}

Exercise #3

Simplify the following:


a7a3= \frac{a^7}{a^3}=

Video Solution

Step-by-Step Solution

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:

cmcn=cmn \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:

a7a3=a73=a4 \frac{a^7}{a^3}=a^{7-3}=a^4 Therefore, the correct answer is option C.

Answer

a4 a^4

Exercise #4

Solve the following:

bybxbzb3= \frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=

Video Solution

Step-by-Step Solution

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

cmcn=cmn \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:

bybxbzb3=byxbz3 \frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=b^{y-x}-b^{z-3} Therefore, the correct answer is option A.

Answer

byxbz3 b^{y-x}-b^{z-3}

Exercise #5

Simplify the following:

a4a6= \frac{a^4}{a^{-6}}=

Video Solution

Step-by-Step Solution

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

cmcn=cmn \frac{c^m}{c^n}=c^{m-n} cmcn=cmn \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:

a4a6=a4(6)=a4+6=a10 \frac{a^4}{a^{-6}}=a^{4-(-6)}=a^{4+6}=a^{10} Therefore, the correct answer is option C.

Answer

a10 a^{10}

Exercise #1

Solve the following exercise:

14a37a3= \frac{14a^{-3}}{7a^{-3}}=

Video Solution

Step-by-Step Solution

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:

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

14a37a3=2a3(3)=2a3+3=2a0 \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:

14a37a3=147a3a3=2a3(3)= \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:

b0=1 b^0=1 Therefore, in the problem we obtain:

2a0=21=2 2a^0=2\cdot1=2 Therefore, the correct answer is option B.

Answer

2 2

Exercise #2

Solve the exercise:

3a22a= \frac{3a^2}{2a}=

Video Solution

Step-by-Step Solution

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:

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

3a22a=32a21=32a1 \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:

3a22a=32a2a=32a21= \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:

b1=b b^1=b We apply it to the problem:

32a1=32a=112a \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.

Answer

112a 1 \frac{1}{2}a

Exercise #3

Solve the following exercise:

3a26a6= \frac{-3a^{-2}}{-6a^{-6}}=

Video Solution

Step-by-Step Solution

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:

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

3a26a6=12a2(6)=12a2+6=12a4 \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:

3a26a6=36a2a6=12a2a6= \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:

12a4 \frac{1}{2}\cdot a^4 Therefore, the correct answer is option C.

Answer

12a4 \frac{1}{2}a^4

Exercise #4

Complete the exercise:

12b44b5= \frac{12b^4}{4b^{-5}}=

Video Solution

Step-by-Step Solution

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:

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

12b44b5=3b4(5)=3b4+5=3b9 \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:

12b44b5=124b4b5=3b4(5)= \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:

3b9 3b^9

Therefore, the correct answer is option D.

Answer

3b9 3b^9

Exercise #5

Solve the exercise:

4a52a3= \frac{4a^5}{2a^3}=

Video Solution

Step-by-Step Solution

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:

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

4a52a3=2a53=2a2 \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:

4a52a3=42a5a3=2a53= \frac{4a^5}{2a^3}=\frac{4}{2}\cdot\frac{a^5}{a^3}=2\cdot a^{5-3}=\ldots We obtained the answer:

2a2 2a^2

Therefore, the correct answer is option A.

Answer

2a2 2a^2

Topics learned in later sections

  1. Rules of Exponentiation
  2. Combining Powers and Roots