Power of a Quotient Rule for Exponents: Applying the formula

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 $13^3$.

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: $\frac{1}{5^7}$.

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 $5^2$.

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.

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 $\frac{x^7}{x^2}$, we need to apply the power of a quotient rule for exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is a nonzero number, and $m$ and $n$ 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 $x$ 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: $7 - 2$.
• Perform the subtraction: $7 - 2 = 5$.
• The resulting exponent of $x$ is 5.

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

The solution to the question is: x^5

To solve the expression $\frac{a^8}{a}$, 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:

• $\frac{b^m}{b^n} = b^{m-n}$

For the given expression:

• $m = 8$
• $n = 1$

Now, applying the formula:

• $\frac{a^8}{a} = a^{8-1} = a^7$

Therefore, the solution to the question is:

a^7

To solve the problem, we apply the quotient rule for exponents. The quotient rule states that when you divide two exponential expressions with the same base, you can subtract the exponent of the denominator from the exponent of the numerator. In mathematical terms:

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

Using the formula mentioned above, let's solve $\frac{b^2}{b^1}$:

• Identify the base for both the numerator and the denominator, which in this case is $b$
• Identify the exponents for the numerator and the denominator: $m = 2$ and $n = 1$
• Subtract the exponent in the denominator from the exponent in the numerator: $2 - 1 = 1$
• Rewrite the expression with the new exponent: $b^{2-1} = b^1$

Thus, $\frac{b^2}{b^1} = b^1$ as per the power of a quotient rule.

The solution to the question is: $b^1$