Power of a Quotient Rule for Exponents: Presenting powers with negative exponents as fractions

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:

$\frac{12^5}{12^8} = 12^{5-8} = 12^{-3}$

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

$12^{-3} = \frac{1}{12^3}$

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

The given expression is: $\frac{\left(4\times3\right)^6}{\left(3\times4\right)^9}$.

We want to simplify this expression using the laws of exponents.

First, notice that the base is the same in both the numerator and the denominator: $4 \times 3$. We can apply the property of exponents that involves a quotient:

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

Thus, we can rewrite the expression as:

$\left(4 \times 3\right)^{6-9}$

Subtract the exponents: $6 - 9 = -3$.

The expression now becomes:

$\left(4 \times 3\right)^{-3}$

To express this with a positive exponent, recall the rule:

$a^{-n} = \frac{1}{a^n}$

Therefore, $\left(4 \times 3\right)^{-3}$ can be written as:

$\frac{1}{\left(4 \times 3\right)^3}$

The solution to the question is: $\frac{1}{\left(4\times3\right)^3}$

Let's solve the given expression step by step:
$\frac{\left(10\times4\right)^2}{\left(10\times4\right)^5}$

Step 1: Use the Power of a Quotient Rule for Exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. Here, $a = 10 \times 4$, $m = 2$, and $n = 5$.

Therefore, apply the rule:

• $\frac{\left(10\times4\right)^2}{\left(10\times4\right)^5} = \left(10\times4\right)^{2-5}$
• $= \left(10\times4\right)^{-3}$

Step 2: Convert the expression with a negative exponent to a fraction:

• We use the rule $a^{-n} = \frac{1}{a^n}$.
• Hence, $\left(10\times4\right)^{-3} = \frac{1}{\left(10\times4\right)^3}$.

The solution to the question is: $\frac{1}{\left(10\times4\right)^3}$

To solve the problem $\frac{\left(6\times7\right)^{13}}{\left(7\times6\right)^{19}}$, we notice that the expressions in both the numerator and the denominator are very similar. Both involve the product of the numbers 6 and 7 raised to some power.

First, we can rewrite the denominator $\left(7 \times 6\right)^{19}$ as $\left(6 \times 7\right)^{19}$. This is possible because the multiplication is commutative, i.e., $a \times b = b \times a$.

Now, the expression becomes:

• $\frac{\left(6\times7\right)^{13}}{\left(6\times7\right)^{19}}$

We can use the rule of exponents, which states that when you divide like bases you subtract the exponents:

• $\frac{a^m}{a^n} = a^{m-n}$

Applying this rule to our expression, we have:

• $\frac{\left(6\times7\right)^{13}}{\left(6\times7\right)^{19}} = (6\times7)^{13-19}$
• $= (6\times7)^{-6}$

Next, we use the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. Therefore,

• $(6\times7)^{-6} = \frac{1}{(6\times7)^6}$

The solution to the question is: $\frac{1}{\left(6\times7\right)^6}$.

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

• Step 1: Apply the quotient rule for exponents
• Step 2: Simplify the exponent
• Step 3: Interpret the negative exponent
• Step 4: Match with the given choices

Let's walk through each step:

Step 1: The quotient rule for exponents states that $\frac{a^m}{a^n} = a^{m-n}$. Here, both the numerator and the denominator are powers with the same base, $11 \times 8$.

Step 2: Apply the rule: $\frac{(11 \times 8)^4}{(11 \times 8)^{11}} = (11 \times 8)^{4 - 11} = (11 \times 8)^{-7}$.

Step 3: The negative exponent $-7$ indicates the reciprocal of the base raised to the positive exponent. Therefore, $(11 \times 8)^{-7} = \frac{1}{(11 \times 8)^7}$.

Step 4: Check the given answer choices:

• Choice 1: $(11\times8)^7$ is incorrect since it does not match our result of the reciprocal.
• Choice 2: $(11\times8)^{-7}$ is correct as it directly matches the simplified expression.
• Choice 3: $\frac{1}{(11\times8)^7}$ is also correct, as it represents the expression with a negative exponent as a fraction.
• Choice 4: "B+C are correct" matches our findings because both choices 2 and 3 are correct representations.

Therefore, the solution to the problem is the choice that states B+C are correct.

To solve this problem, follow these steps:

• Step 1: Simplify the expression using exponent rules.
• Step 2: Compare with the given answer choices.

Step 1: Simplify the expression $\frac{(8 \times 2)^3}{(2 \times 8)^7}$.
Note that the bases in both the numerator and denominator are identical: $8 \times 2 = 16$. So the expression can be rewritten as:

$\frac{16^3}{16^7}$.

Using the exponent division rule, $\frac{a^m}{a^n} = a^{m-n}$, simplify the expression:

$16^{3-7} = 16^{-4}$.

According to the negative exponent rule, $a^{-n} = \frac{1}{a^n}$, this becomes:

$\frac{1}{16^4}$.

Step 2: Compare the simplification $\frac{1}{16^4}$ to the answer choices:

• Choice 1: $(8 \times 2)^{-4}$: Indicates $16^{-4}$, which is correct.
• Choice 2: $\frac{1}{(8 \times 2)^4}$: Equivalent to $\frac{1}{16^4}$, which is also correct.
• Choice 3: $\frac{1}{(8 \times 2)^{-4}}$: Implies $16^4$, which is incorrect.
• Choice 4: $a'+b'$ are correct: Is correct since both A and B represent the same solution.

Therefore, the correct choice is Choice 4: a'+b' are correct.

I'm confident in this solution as it accurately applies the rules of exponents. All recalculations confirm the analysis and answers provided.