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

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: $5^0$

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 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 $\frac{60^{60}}{60^{42}}$, 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 $\frac{13^{17}}{13^{14}}$, 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 $13^3$.

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 $\frac{17^{16}}{17^{20}}$, 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 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 $8^{-5}$.

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: $5^{-5}$

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 the expression $\frac{8}{8^4}$, we will simplify it using the exponent rule for dividing powers with the same base:

• Step 1: Identify the expression as $\frac{8^1}{8^4}$. Both terms have base 8.
• Step 2: Apply the formula $\frac{a^m}{a^n} = a^{m-n}$. Here, $m = 1$ and $n = 4$.
• Step 3: Perform the subtraction in the exponent: $8^{1-4}$.

Now, calculating the exponent:

$8^{1-4} = 8^{-3}$.

We know that a negative exponent indicates the reciprocal, so:

$8^{-3} = \frac{1}{8^3}$.

Thus, the simplified expression is $\frac{1}{8^3}$.

Based on the choices given, the correct option is:

• $\frac{1}{8^3}$ (Choice 1): This matches our simplified expression.
• $8^3$ (Choice 2): Incorrect, as it does not simplify the division.
• $8^{-4}$ (Choice 3): Incorrect, because it incorrectly represents the situation.
• $8^4$ (Choice 4): Incorrect, as it doesn't simplify the expression.

Therefore, the solution to the problem is: $\frac{1}{8^3}$.

I am confident in the correctness of this solution.

To solve this problem, we'll follow these steps:

• Step 1: Apply the quotient rule for exponents
• Step 2: Simplify the resulting expression
• Step 3: Compare the simplified form with the given choices

Now, let's work through each step:
Step 1: Given the expression $\frac{7^{10}}{7^{13}}$, use the quotient rule for exponents, which states $\frac{a^m}{a^n} = a^{m-n}$.
Step 2: Apply this rule to get $7^{10-13} = 7^{-3}$.
Step 3: Rewrite $7^{-3}$ using the rule for negative exponents, which is $a^{-n} = \frac{1}{a^n}$. Therefore, $7^{-3} = \frac{1}{7^3}$.

Comparing with the provided answer choices, the correct choice is:

• Choice 2: $\frac{1}{7^3}$

Therefore, the solution to the problem is $\frac{1}{7^3}$, confirming the correctness of the derived expression and matching the provided answer.

Let's solve the expression $\frac{2^2}{2^6}$ using the rules of exponents. Specifically, we'll use the Power of a Quotient Rule for Exponents which states that $\frac{a^m}{a^n} = a^{m-n}$.

• First, identify the base, which is 2, and the exponents. According to the rule, we subtract the exponent in the denominator from the exponent in the numerator.
• In our case, the exponents are 2 (in the numerator) and 6 (in the denominator).
• Subtract the exponent in the denominator from the exponent in the numerator: $2 - 6 = -4$. This gives us $2^{-4}$.
• According to the rule of negative exponents, $a^{-n} = \frac{1}{a^n}$, so we rewrite $2^{-4}$ as $\frac{1}{2^4}$.

Therefore, the expression $\frac{2^2}{2^6}$ simplifies to $\frac{1}{2^4}$.

The solution to the question is: $\frac{1}{2^4}$

To simplify the expression $\frac{12^5}{12^8}$, we'll follow these steps:

• Step 1: Apply the quotient rule for exponents.
• Step 2: Simplify and interpret the result using negative exponents if necessary.

Let's work through each step:

Step 1: Apply the quotient rule for exponents.
We are given the expression $\frac{12^5}{12^8}$. According to the quotient rule for exponents, $\frac{a^m}{a^n} = a^{m-n}$, so we have:

Step 2: Simplify and interpret.
The result $12^{-3}$ can be expressed using the concept of negative exponents $a^{-n} = \frac{1}{a^n}$:

Therefore, both expressions $12^{-3}$ and $\frac{1}{12^3}$ are equivalent.

Matching with the provided choices:
- Choice 1: $12^{-3}$ - This matches our first result.
- Choice 2: $\frac{1}{12^3}$ - This matches our interpretation of the negative exponent.

Choice 4 states: "a'+b' are correct," which refers to both expressions being correct representations. Therefore, the correct answer is "a'+b' are correct."

To solve this problem, we'll use the quotient rule for exponents:

The formula to use is $\frac{a^m}{a^n} = a^{m-n}$, applicable when the bases are the same.

Let's apply this step-by-step:

• Step 1: We have the expression $\frac{5^6}{5^{13}}$. Identify $m = 6$ and $n = 13$ with base $5$.
• Step 2: Apply the formula $\frac{5^6}{5^{13}} = 5^{6-13}$.
• Step 3: Simplify the exponent: $6 - 13 = -7$, so $5^{6-13} = 5^{-7}$.
• Step 4: Recognize that a negative exponent means a reciprocal: $5^{-7} = \frac{1}{5^7}$.

Therefore, the solution to the problem is $\frac{1}{5^7}$.

Now, considering the answer choices:

• Choice 1: $5^7$, not correct as it doesn't reflect the negative exponent.
• Choice 2: $\frac{1}{5^7}$, correct since it matches our simplified solution.
• Choice 3: $5^{19}$, incorrect; this would imply adding exponents.
• Choice 4: a'+b' are correct. This is not relevant to our examined choices.

Thus, the correct option is Choice 2: $\frac{1}{5^7}$.