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.

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

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

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

This simplification leads us to:

$13^{17-14} = 13^3$

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

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.

The given expression is $\frac{8^{16}}{8^8}$. 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:

• $\frac{a^m}{a^n} = a^{m-n}$

In this problem, the base $8$ 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:

• $16 - 8 = 8$

Therefore, the simplified form of the given expression is:

• $8^8$

Thus, the answer is $8^8$.

To simplify the expression $\frac{8^4}{8^9}$, we apply the rule of exponents for division:

• The quotient rule for exponents is $\frac{a^m}{a^n} = a^{m-n}$.

Since both the numerator and the denominator have the same base (8), we can apply this rule directly:

$\frac{8^4}{8^9} = 8^{4-9}$

Thus, the resulting expression is $8^{-5}$.

Reviewing the choices given:

• Choice 1: $8^{9-4}$ which equals $8^5$, is incorrectly stating the subtraction order.
• Choice 2: $8^{\frac{4}{9}}$ is incorrect, as it represents a different operation (taking the root) rather than division of exponents.
• Choice 3: $8^{4-9}$ is correct, as it correctly applies the quotient rule for exponents.
• Choice 4: $8^{4\times9}$ suggests multiplication of exponents, not applicable here.

Therefore, the correct answer is Choice 3: $8^{4-9}$, which simplifies to $8^{-5}$.

To solve the expression $\frac{17^{16}}{17^{20}}$, we can apply the Power of a Quotient Rule for Exponents. This rule states that when you divide two exponents with the same base, you can subtract the exponents to simplify the expression.

The given expression is:

$\frac{17^{16}}{17^{20}}$

According to the Quotient Rule for Exponents, this expression can be simplified as:

$17^{16-20}$

Here's the step-by-step breakdown:

• The base of both the numerator and the denominator is the same, that is, 17.
• According to the rule, subtract the exponent in the denominator from the exponent in the numerator: $16 - 20$.
• This gives us the exponent: $-4$.

So, the simplified expression is:

$17^{-4}$

However, as requested, we should express this as:

$17^{16-20}$

The solution to the question is:

$17^{16-20}$

To solve the expression $\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.

• $a^m \div a^n = a^{m-n}$

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

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

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

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

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

The solution to the question is: 25^7

To solve the expression $\frac{5^7}{5^{10}}$, we need to apply the Power of a Quotient Rule for Exponents. This rule states that when dividing like bases, you subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$.

In this particular case, the base is 5, and the exponents are 7 and 10. Using the rule, we subtract the exponent in the denominator from the exponent in the numerator:

• Numerator exponent = 7
• Denominator exponent = 10

The solution to the question is: $5^{7-10}$.

To solve the expression $\frac{60^{60}}{60^{42}}$, 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 $a$, and integers $m$ and $n$, $\frac{a^m}{a^n} = a^{m-n}$.

Applying this rule to our problem:

• We have the same base:$60$.

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

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

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

To solve the expression $\frac{9^{15}}{9^{10}}$, we will use the Power of a Quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. 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:
  $9^{15-10}$
• Calculate the result of the subtraction:
  $15 - 10 = 5$
• Thus, the simplified form of the expression is:
  $9^5$

The solution to the question is: $9^5$

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:

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

For our problem, the expression is:

$\frac{5^9}{5^9}$

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

$\frac{5^9}{5^9} = 5^{9-9}$

Calculating the exponent:

$9 - 9 = 0$

So the expression becomes:

$5^0$

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, $5^0$ is the correct answer.

The solution to the question is: $5^0$

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 $5^2$.

We need to simplify the expression $\frac{5^3}{5^8}$ 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:

$\frac{a^m}{a^n} = a^{m-n}$.

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

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

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

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