Power of a Quotient Rule for Exponents: Calculating powers with negative exponents

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: $9^5$

We start by analyzing the expression: $\frac{\left(16\times7\right)^6}{\left(16\times7\right)^8}$.

This expression is a perfect candidate for applying the Power of a Quotient Rule for Exponents, which states:

$\frac{a^m}{a^n} = a^{m-n}$, where $a$ is a nonzero number, and $m$ and $n$ are integers.

In our case, $a = 16 \times 7$, $m = 6$, and $n = 8$.

Applying the rule, we subtract the exponents of the base $16 \times 7$:

$\frac{\left(16\times7\right)^6}{\left(16\times7\right)^8} = \left(16\times7\right)^{6-8}$.

Now, simplify the exponent:

$6 - 8 = -2$

Thus, the expression simplifies to:

$\left(16\times7\right)^{-2}$.

However, comparing with the provided solution, it shows $\left(16\times7\right)^{6-8}$, which is the form before the numerical simplification of the exponent.

The solution to the question is: $\left(16\times7\right)^{6-8}$.

Let's simplify the expression $\frac{\left(15\times4\right)^3}{\left(4\times15\right)^9}$:

We start by recognizing that both the numerator and the denominator share the same base: $(15 \times 4)$. Therefore, we have a quotient of powers with the same base:

$\frac{\left(15\times4\right)^3}{\left(15\times4\right)^9}$

According to the rules of exponents, when dividing like bases, we subtract the exponents:

$(15 \times 4)^{3 - 9}$

Subtracting the exponents, we have:

$(15 \times 4)^{-6}$

This matches with one of the choices:

• Choice 1: $(15\times4)^{3-9}$ is correct, as it simplifies to $(15\times4)^{-6}$.
• Choice 2 and Choice 3 involve incorrect operations on exponents (multiplication and addition respectively).
• Choice 4 reverses the subtraction order, which results differently in base powers than what is needed.

Therefore, the correct answer to the problem is:
$\left(15\times4\right)^{3-9}$.

Let's solve the given expression by applying the rules of exponents. The expression given is:
$\frac{\left(2\times7\right)^4}{\left(2\times7\right)^7}$

We know the rule for dividing powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$.
In this case, the base is $2 \times 7$, and we have the exponent 4 in the numerator and 7 in the denominator.

Applying the rule, we subtract the exponent in the denominator from the exponent in the numerator:

• $\frac{\left(2\times7\right)^4}{\left(2\times7\right)^7} = \left(2\times7\right)^{4-7}$.

Now simplify the exponent:

• $4 - 7 = -3$

Thus, the expression becomes:
$\left(2\times7\right)^{-3}$.

The solution to the question is: $\left(2\times7\right)^{-3}$.

To solve this problem, we will use the power of a quotient rule for exponents, which states: $\frac{a^m}{a^n} = a^{m-n}$.

Here's a step-by-step solution:

• Step 1: Identify the base and exponents in the problem.
• Step 2: Apply the exponent rules.
• Step 3: Simplify the expression by calculating the difference in exponents.

Now, let's apply these steps in detail:

Step 1: We have the expression $\frac{(12 \times 4)^4}{(4 \times 12)^{11}}$. Here, the base is $(12 \times 4)$, and the exponents are 4 in the numerator and 11 in the denominator.

Step 2: Combine the terms using the commutative and associative properties of multiplication. Notice that the terms $12 \times 4$ are identical in both the numerator and denominator. So, simplify using these instances: $(12 \times 4)^4$ in the numerator and $(12 \times 4)^{11}$ in the denominator.

Step 3: Apply the power of a quotient rule:
$\frac{(12 \times 4)^4}{(12 \times 4)^{11}} = (12 \times 4)^{4-11} = (12 \times 4)^{-7}$

This means the simplified expression is $(12 \times 4)^{-7}$.

Therefore, the correct answer among the provided choices is $\left(12\times4\right)^{-7}$, which corresponds to choice 1.

To solve the given problem, we need to carefully apply the rules of exponents, particularly the power of a quotient rule, which states $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$, and the rule for negative exponents, which is $a^{-m} = \frac{1}{a^m}$.

Given equation: $\frac{(3\times13)^{14}}{(13\times3)^{20}}$

First, notice that in both the numerator and the denominator, the terms are the same, just written in reverse order:

• Numerator: $(3\times13)^{14}$

• Denominator: $(13\times3)^{20}$

Since multiplication is commutative, we have:

• $(3\times13) = (13\times3)$

Therefore, the expression simplifies to:

• $\frac{(3\times13)^{14}}{(3\times13)^{20}}$

Since the bases are now identical, we can apply the rule $\frac{a^m}{a^n} = a^{m-n}$:

The exponent in the numerator is 14 and in the denominator is 20, giving us:

• $(3\times13)^{14-20}$

Calculate the subtraction in the exponent:

• $14 - 20 = -6$

Ultimately, the expression simplifies to:

• $(3\times13)^{-6}$

Therefore, the solution to the problem is: $(3\times13)^{-6}$