Examples with solutions for Power of a Quotient Rule for Exponents: Applying the formula

Exercise #1

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 #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

9993= \frac{9^9}{9^3}=

Video Solution

Step-by-Step Solution

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:

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

9993=993=96 \frac{9^9}{9^3}=9^{9-3}=9^6 Therefore, the correct answer is b.

Answer

96 9^6

Exercise #4

Insert the corresponding expression:

5959 \frac{5^9}{5^9}

Video Solution

Step-by-Step Solution

To solve this problem, we apply the Power of a Quotient Rule for exponents. This rule is applicable when both the numerator and the denominator of a fraction have the same base. The rule states:


aman=amn \frac{a^m}{a^n} = a^{m-n}


For our problem, the expression is:


5959 \frac{5^9}{5^9}


In this expression, the base a a is 5 5 , m m is 9 9 , and n n is also 9 9 . We apply the rule as follows:


5959=599 \frac{5^9}{5^9} = 5^{9-9}


Calculating the exponent:


99=0 9 - 9 = 0


So the expression becomes:


50 5^0


Any number raised to the power of 0 is 1, but in this context, we are simply reducing the original expression to its simplest form. Therefore, 50 5^0 is the correct answer.


The solution to the question is: 50 5^0

Answer

50 5^0

Exercise #5

Insert the corresponding expression:

81688= \frac{8^{16}}{8^8}=

Video Solution

Step-by-Step Solution

The given expression is 81688 \frac{8^{16}}{8^8} . To solve this, we apply the Power of a Quotient Rule for Exponents.

This rule states that when dividing two exponential expressions with the same base, we subtract the exponent of the denominator from the exponent of the numerator. Mathematically, it can be expressed as:

  • aman=amn \frac{a^m}{a^n} = a^{m-n}

In this problem, the base 8 8 is the same in both the numerator and the denominator, so we can apply this rule.

Subtract the exponent of the denominator from the exponent of the numerator:

  • 168=8 16 - 8 = 8

Therefore, the simplified form of the given expression is:

  • 88 8^8

Thus, the answer is 88 8^8 .

Answer

88 8^8

Exercise #6

Insert the corresponding expression:

259252= \frac{25^9}{25^2}=

Video Solution

Step-by-Step Solution

To solve the expression 259252 \frac{25^9}{25^2} , we will use the Power of a Quotient Rule for Exponents. According to this rule, when dividing like bases, we subtract the exponents.


  • am÷an=amn a^m \div a^n = a^{m-n}


In the given expression, the base 25 25 is the same for both the numerator and the denominator. Therefore, we can apply the rule as follows:


  • Identify the exponents: m=9 m = 9 and n=2 n = 2 .

  • Subtract the exponents: 92=7 9 - 2 = 7 .

  • Write the result as a single power of the base: 257 25^7 .


Thus, the expression 259252 \frac{25^9}{25^2} simplifies to 257 25^7 .


The solution to the question is: 25^7

Answer

257 25^7

Exercise #7

Insert the corresponding expression:

60606042= \frac{60^{60}}{60^{42}}=

Video Solution

Step-by-Step Solution

To solve the expression 60606042 \frac{60^{60}}{60^{42}} , we need to apply the Power of a Quotient Rule for Exponents. This rule states that when dividing like bases, we subtract the exponents. In mathematical terms, for any non-zero number a a , and integers m m and n n , aman=amn \frac{a^m}{a^n} = a^{m-n} .

Applying this rule to our problem:

  • We have the same base:60 60 .

  • We subtract the exponent in the denominator from the exponent in the numerator: 606042 60^{60-42} .

  • This simplifies the expression to 6018 60^{18} .

Therefore, the solution to the question is: 6018 60^{18} .

Answer

6018 60^{18}

Exercise #8

Insert the corresponding expression:

13171314= \frac{13^{17}}{13^{14}}=

Video Solution

Step-by-Step Solution

To solve the expression 13171314 \frac{13^{17}}{13^{14}} , we use the Power of a Quotient Rule for Exponents. This rule states that aman=amn \frac{a^m}{a^n} = a^{m-n} , where a a is a non-zero number, and m m and n n are integers.


In the given expression, a=13 a = 13 , m=17 m = 17 , and n=14 n = 14 . Applying the power of a quotient rule, we perform the following calculation:


Subtract the exponent in the denominator from the exponent in the numerator: 1714=3 17 - 14 = 3 .


This simplification leads us to:

131714=133 13^{17-14} = 13^3


Therefore, the final simplified expression is 133 13^3 .

Answer

133 13^3

Exercise #9

Insert the corresponding expression:

5358= \frac{5^3}{5^8}=

Video Solution

Step-by-Step Solution

We need to simplify the expression 5358 \frac{5^3}{5^8} using the rules of exponents. Specifically, we will use the power of a quotient rule for exponents which states that when you divide like bases you subtract the exponents:

aman=amn \frac{a^m}{a^n} = a^{m-n} .

Here, the base is 5, the exponent in the numerator is 3, and the exponent in the denominator is 8.

  • Apply the rule: 538 5^{3-8}
  • Subtract the exponents: 55 5^{-5} .

Therefore, the simplified expression is 55 5^{-5} .

The solution to the question is: 55 5^{-5}

Answer

55 5^{-5}

Exercise #10

Insert the corresponding expression:

915910= \frac{9^{15}}{9^{10}}=

Video Solution

Step-by-Step Solution

To solve the expression 915910 \frac{9^{15}}{9^{10}} , we will use the Power of a Quotient rule for exponents, which states that aman=amn \frac{a^m}{a^n} = a^{m-n} . This rule applies when both the numerator and the denominator have the same base.

In our problem, both the numerator and the denominator have the base 9, hence we can apply the rule:

  • Identify the exponents: The exponent in the numerator is 15, and the exponent in the denominator is 10.
  • Apply the Power of a Quotient rule by subtracting the exponent of the denominator from the exponent of the numerator:
    91510 9^{15-10}
  • Calculate the result of the subtraction:
    1510=5 15 - 10 = 5
  • Thus, the simplified form of the expression is:
    95 9^5

The solution to the question is: 95 9^5

Answer

95 9^5

Exercise #11

3532= \frac{3^5}{3^2}=

Video Solution

Step-by-Step Solution

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

Here, we have 3532=352 \frac{3^5}{3^2} = 3^{5-2}

Simplifying, we get 33 3^3

Answer

33 3^3

Exercise #12

5654= \frac{5^6}{5^4}=

Video Solution

Step-by-Step Solution

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

Here, we have 5654=564 \frac{5^6}{5^4} = 5^{6-4} . Simplifying, we get 52 5^2 .

Answer

52 5^2

Exercise #13

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 #14

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 #15

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 #16

Choose the expression that is equal to the following:

a5:a4 a^5:a^4

Video Solution

Step-by-Step Solution

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

a5a4 \frac{a^5}{a^4} Then we'll recall the law of exponents for division between terms with equal bases:

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

a5a4=a54=a1=a \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:

X1=X X^1=X We got that: a5a4=a \frac{a^5}{a^4}=a therefore the correct answer is a.

Answer

a a

Exercise #17

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 #18

Solve the following exercise

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

Video Solution

Step-by-Step Solution

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:

cmcn=cmn \frac{c^m}{c^n}=c^{m-n} We apply the previously mentioned property to 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 #19

Insert the corresponding expression:

x7x2= \frac{x^7}{x^2}=

Video Solution

Step-by-Step Solution

To solve the expression x7x2 \frac{x^7}{x^2} , we need to apply the power of a quotient rule for exponents. This rule states that aman=amn \frac{a^m}{a^n} = a^{m-n} , where a a is a nonzero number, and m m and n n are integers. This rule allows us to subtract the exponent in the denominator from the exponent in the numerator, given that the bases are the same.

Here's a step-by-step breakdown of applying the formula:

  • Identify the base x x which is common in both numerator and denominator.
  • Look at the exponents: the numerator has an exponent of 7 and the denominator has an exponent of 2.
  • According to the power of a quotient rule, subtract the exponent in the denominator from the exponent in the numerator: 72 7 - 2 .
  • Perform the subtraction: 72=5 7 - 2 = 5 .
  • The resulting exponent of x x is 5.

Therefore, x7x2 \frac{x^7}{x^2} simplifies to x5 x^5 .

The solution to the question is: x^5

Answer

x5 x^5

Exercise #20

Insert the corresponding expression:

a8a= \frac{a^8}{a}=

Video Solution

Step-by-Step Solution

To solve the expression a8a \frac{a^8}{a} , we can use the Power of a Quotient Rule for Exponents. According to this rule, when dividing like bases, we subtract the exponents.


The general formula for this is:


  • bmbn=bmn \frac{b^m}{b^n} = b^{m-n}

For the given expression:


  • m=8m = 8
  • n=1n = 1

Now, applying the formula:


  • a8a=a81=a7 \frac{a^8}{a} = a^{8-1} = a^7

Therefore, the solution to the question is:


a^7

Answer

a7 a^7