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: $9^5$

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: $b^1$

The given expression is $\frac{y^7}{y^2}$.

To simplify this expression, we apply the Power of a Quotient Rule for exponents. This rule states that when you divide two expressions with the same base, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

In this case, the base $y$ is the same for both the numerator and the denominator.

• The exponent in the numerator is 7.
• The exponent in the denominator is 2.

Thus, following the rule, we subtract the exponent of the denominator from the exponent of the numerator:

$y^{7-2} = y^5$

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

To solve the expression $\frac{x^9}{x^2}$, we will 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 non-zero number, and $m$ and $n$ are integers. In our case, $a$ is $x$, $m$ is 9, and $n$ is 2.

Now, apply the formula:

• The expression can be rewritten as $x^{9-2}$.
• Calculate the exponent: $9 - 2 = 7$.

Substitute back to get the simplified expression: $x^7$.

The solution to the question is: $x^7$

To solve the question, let's apply the Power of a Quotient Rule for Exponents. This rule states that when dividing like bases with exponents, you can subtract the exponent of the denominator from the exponent of the numerator. Mathematically, this is expressed as:

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

In our case, the base $b$ is 3, the exponent $m$ for the numerator is $2a$, and the exponent $n$ for the denominator is $a$. Thus, we can substitute these into the formula:

$\frac{3^{2a}}{3^a} = 3^{2a-a}$

Now, simplify the exponent:

$3^{2a-a} = 3^{a}$

Therefore, the expression simplifies to:

$3^a$

The solution to the question is: $3^a$

To solve the given expression $\frac{6^{4x}}{6^{x+1}}$, we must apply the Power of a Quotient Rule for Exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$.

Using this rule, the given expression can be rewritten as follows:

• The numerator is $6^{4x}$.
• The denominator is $6^{x+1}$.

Apply the Power of a Quotient Rule:

$\frac{6^{4x}}{6^{x+1}} = 6^{4x - (x + 1)}$

We need to simplify the exponent by performing the subtraction $4x - (x + 1)$:

Step 1: Distribute the subtraction sign to the terms inside the parenthesis:

• $4x - x - 1$

Step 2: Combine like terms:

• $3x - 1$

The expression simplifies to:

$6^{3x-1}$

Therefore, the solution to the question is: $6^{3x-1}$.

To solve the problem $\frac{11^{5a}}{11^{a-4}}$, we need to use the Power of a Quotient Rule for exponents, which states that $\frac{b^m}{b^n} = b^{m-n}$.

Let's apply this rule to the given expression:

• The base is $11$, which is the same for both the numerator and the denominator.
• The exponent in the numerator is $5a$.
• The exponent in the denominator is $a - 4$.

According to the formula $\frac{b^m}{b^n} = b^{m-n}$, we can subtract the exponent in the denominator from the exponent in the numerator:

$5a - (a - 4) = 5a - a + 4$.

This simplifies to $4a + 4$.

Therefore, $\frac{11^{5a}}{11^{a-4}} = 11^{4a + 4}$.

The correct answer provided was $11^{4a-4}$.

Therefore, the final expression we arrived at using the Power of a Quotient Rule is: $11^{4a + 4}$.

I couldn't get to the shown answer.