Power of a Quotient Rule for Exponents: Number of terms

To solve the problem, we first need to apply the exponent rules, specifically focusing on the "Power of a Quotient" rule. The given expression is:

$\frac{\left(10\times2\right)^{20}}{\left(2\times10\right)^7}$

We can notice that both the numerator and the denominator have the same base, which is $(10 \times 2) \ or \ (2 \times 10)$. Hence, let's simplify the base:

• $a = 10 \times 2 = 20$

Thus, both the numerator and the denominator can be rewritten with the base $a$:

• $a^{20}$ for the numerator

• $a^{7}$ for the denominator

Now, using the "Power of a Quotient" rule:

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

We apply this rule to our expression:

$\frac{a^{20}}{a^7} = a^{20-7}$

This simplifies to:

$a^{13}$

Substituting back the value of $a$:

$\left(2 \times 10\right)^{13}$

However, let's check the solution form given in the problem:

The solution hinted at is:

$\left(2 \times 10\right)^{20-7}$

Indeed, it verifies our calculation that the expression simplifies to $\left(2 \times 10\right)^{13}$.

The solution to the question is: $\left(2 \times 10\right)^{13}$

We begin by analyzing the given expression: $\frac{\left(9\times5\right)^{12}}{\left(5\times9\right)^6}$. Using the property of exponents known as the Power of a Quotient Rule, we can rewrite this expression.
This rule states that $\frac{a^m}{a^n} = a^{m-n}$. Here, both the numerator and the denominator have the same base, $9\times5$ or equivalently $5\times9$, therefore we can apply this rule.

Let's apply the Power of a Quotient Rule:

• Identify the base, which is $9\times5$.

• Subtract the exponent in the denominator from the exponent in the numerator: $12 - 6$.

Thus, the expression simplifies to $\left(9\times5\right)^{12-6}$.

The solution to the question is: $\left(9\times5\right)^{12-6}$.

We need to simplify the expression: $\frac{\left(3\times6\right)^{10}}{\left(3\times6\right)^7}$.

According to the Power of a Quotient Rule for Exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$, we can simplify any fraction where the numerator and the denominator have the same base and different exponents by subtracting their exponents.

In our case, the common base is $3\times6$. Let's apply the rule:

• The exponent in the numerator is 10.
• The exponent in the denominator is 7.

So, according to the rule, we subtract the exponent in the denominator from the exponent in the numerator:

$(3\times6)^{10-7}$.

Thus, the expression simplifies to $\left(3\times6\right)^{10-7}$.

To solve the given expression $\frac{\left(7\times2\right)^{9}}{\left(2\times7\right)^2}$, we will apply the Power of a Quotient Rule for Exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$.

The base of the exponents in both the numerator and the denominator is the same, $7 \times 2$ or equivalently $2 \times 7$.

1. First, note that the structure is $\frac{(7\times2)^9}{(7\times2)^2}$.

2. Using the Power of a Quotient Rule: $\frac{(7\times2)^9}{(7\times2)^2} = (7\times2)^{9-2}$$$

3. Simplify the expression in the exponent: $9 - 2 = 7$

4. Therefore, the simplified expression is \$(7\times2)^7$

The solution to the question is: $(7\times2)^{9-2}$

We start with the given expression:
$\frac{\left(4\times5\right)^{8}}{\left(4\times5\right)^4}$

According to the power of a quotient rule for exponents, we can simplify an expression of the form $\frac{a^m}{a^n}$ as $a^{m-n}$.
This rule states that when we divide two exponents with the same base, we subtract the exponents.

Applying this rule to our expression, we have:

• Base: $4 \times 5$
• Exponent in the numerator: $8$
• Exponent in the denominator: $4$

Thus, we subtract the exponents in the quotient:

$(4\times5)^{8-4}$

Simplifying the exponent:

$(4\times5)^{4}$

Therefore, the expression simplifies to:
$(4\times5)^{8-4}$.

The solution to the question is $\left(4\times5\right)^{8-4}$.

Let's solve the given expression step by step by using the power of a quotient rule for exponents. The rule states that $\frac{a^n}{a^m} = a^{n-m}$, where $a$ is any non-zero number, and $n$ and $m$ are integers.

Given the expression: $\frac{\left(2\times3\right)^{6}}{\left(2\times3\right)^3}$

• First, apply the power of a quotient rule for exponents formula: $\frac{\left(2\times3\right)^{6}}{\left(2\times3\right)^3} = \left(2\times3\right)^{6-3}$.

• The exponent in the numerator is 6, and the exponent in the denominator is 3.

• Subtract the exponent in the denominator from the exponent in the numerator: $6 - 3 = 3$.

• Thus, the expression simplifies to: $\left(2\times3\right)^3$.

The solution to the question is: $\left(2\times3\right)^{6-3}$

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

• Step 1: Identify the expression's base and exponents.
• Step 2: Apply the quotient rule for exponents.
• Step 3: Simplify the expression.

Now, let's work through each step:

Step 1: The given expression is $\frac{(16 \times 5)^{25}}{(16 \times 5)^{21}}$, where the base is $(16 \times 5)$ and the exponents are 25 and 21.

Step 2: Applying the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$, we subtract the exponent of the denominator from the exponent of the numerator. Thus, we have:

$(16 \times 5)^{25-21}$

Step 3: Simplifying the exponents results in:

$(16 \times 5)^4$

Therefore, the correct choice is the simplified expression:

$(16 \times 5)^{25-21}$

Upon reviewing the provided choices:

• Choice 1: $(16 \times 5)^{25 \times 21}$ is incorrect because it indicates multiplication of exponents, not the subtraction needed.
• Choice 2: $(16 \times 5)^{25-21}$ is correct, reflecting the exponent subtraction.
• Choice 3: $(16 \times 5)^{25+21}$ is incorrect as it applies the wrong rule (addition instead of subtraction).
• Choice 4: $(16 \times 5)^{\frac{25}{21}}$ is incorrect as the quotient rule for exponents requires subtraction, not division.

Thus, choice 2 is the correct answer.

The given expression is $\frac{\left(15\times2\right)^{17}}{\left(2\times15\right)^{13}}$. To simplify
using the rule of exponents known as the Power of a Quotient Rule, which states

When you divide like bases you subtract the exponents:

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

First, notice that both the numerator and denominator have the base $15 \times 2$. Therefore, we can simplify by subtracting the exponents in the numerator and the denominator:

• Numerator's exponent: 17
• Denominator's exponent: 13

We are given the expression: $\frac{\left(8\times7\right)^{15}}{\left(8\times7\right)^3}$

To solve this, we can use the Power of a Quotient Rule for Exponents. This rule states that for any non-zero numbers $a$ and $b$, and any integers $m$ and $n$, the expression:

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

can be simplified by subtracting the exponent in the denominator from the exponent in the numerator.

Using the Power of a Quotient Rule, let's apply it to our expression:

Given: $\frac{\left(8\times7\right)^{15}}{\left(8\times7\right)^3}$

According to the rule: $\left(8\times7\right)^{15-3}$

So, the simplified expression is: $\left(8\times7\right)^{12}$

Thus, the correct simplified expression is: $\left(8\times7\right)^{15-3}$

Let's solve the given mathematical expression step by step using the rules of exponents.

• We start with the expression: $\frac{\left(11\times12\right)^{30}}{\left(11\times12\right)^{30}}$.

• According to the rules of exponents, specifically the quotient rule, which states that when you divide powers with the same base, you subtract their exponents: $\frac{a^m}{a^n} = a^{m-n}$.

• Applying this rule to the expression, since the base $11 \times 12$ is the same in both the numerator and the denominator, we subtract the exponents:

  • The numerator is $\left(11\times12\right)^{30}$ and the denominator is $\left(11\times12\right)^{30}$.

  • Therefore, $\frac{\left(11\times12\right)^{30}}{\left(11\times12\right)^{30}} = \left(11\times12\right)^{30-30}$.

• Simplifying further, we have:

  • $\left(11\times12\right)^{30-30} = \left(11\times12\right)^{0}$.

  • Any non-zero number raised to the power of 0 is 1. However, here the expression is left in the form of an exponent as requested.

The solution to the question is: $\left(11\times12\right)^{30-30}$

To solve the expression $\frac{\left(12\times6\right)^{20}}{\left(6\times12\right)^4}$, we will use the Power of a Quotient Rule for Exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$.

First, let's simplify the expression inside the parentheses.

The numerator is: $(12 \times 6)^{20}$ and the denominator is: $(6 \times 12)^4$.

Notice that $12 \times 6 = 72$. Therefore, our expression simplifies to:

$\frac{72^{20}}{72^4}$

Applying the Power of a Quotient Rule, we have:

$72^{20-4} = 72^{16}$

Thus, the expression simplifies to $72^{16}$.

The solution to the question is: $72^{16}$. A'+C' are correct.

To solve the given expression $\frac{\left(25\times2\right)^{16}}{\left(25\times2\right)^5}$, we need to apply the Power of a Quotient Rule for Exponents. This rule states that when you have the same base, you can subtract the exponent of the denominator from the exponent of the numerator. The general formula is:

• $\frac{a^m}{a^n} = a^{m-n}$

Here, the base $a$ is $25 \times 2$, the numerator's exponent $m$ is 16, and the denominator's exponent $n$ is 5.

Now, apply the Power of a Quotient Rule:

$\frac{\left(25\times2\right)^{16}}{\left(25\times2\right)^5} = \left(25\times2\right)^{16-5}$

Subtract the exponents:

$\left(25\times2\right)^{16-5} = \left(25\times2\right)^{11}$

Therefore, the simplified expression is:

$\left(25\times2\right)^{11}$

The solution to the question is: $\left(25\times2\right)^{11}$

Let's start solving this equation step by step. The problem provided is:

$\frac{\left(17\times3\right)^{17}}{\left(17\times3\right)^{11}}=$

This problem involves the Power of a Quotient Rule for Exponents, which states:

• If you have a quotient of terms with the same base, you can subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

The terms in our problem already have the same base $(17\times3)$. Therefore, we apply the rule directly:

$\frac{\left(17\times3\right)^{17}}{\left(17\times3\right)^{11}} = \left( 17 \times 3 \right)^{17-11}$

Simplifying the exponent gives

$17 - 11 = 6$

Thus, the expression simplifies to:

$\left(17 \times 3\right)^6$

Therefore, the solution to the question is:

$\left(17 \times 3\right)^6$

The question requires us to simplify the given expression using the laws of exponents, specifically the Power of a Quotient Rule for Exponents. The given expression is:

$\frac{\left(7\times13\right)^{13}}{\left(13\times7\right)^{17}}$

We can rewrite the expression inside both the numerator and the denominator to express them more clearly:

$\left(7\times13\right)^{13} = (91)^{13}$ and $\left(13\times7\right)^{17} = (91)^{17}$

The expression now looks like this:

$\frac{(91)^{13}}{(91)^{17}}$

According to the properties of exponents, specifically the rule for dividing same bases, we subtract the exponents:

$(91)^{13-17} = (91)^{-4}$

The expression now simplified is:

$\frac{1}{(91)^4}$

Therefore, we see that the simplified answer does not directly correspond to the given answer of "a' + b' = c'." It seems there might be a discrepancy in the final simplification or understanding, as we derived:

The solution to the question is: $\frac{1}{(91)^4}$

I couldn't get to the shown answer, "a'+b' are correct."

The given expression is $\frac{\left(4\times7\right)^{12}}{\left(4\times7\right)^5}$.
We are asked to simplify this expression using the Power of a Quotient Rule for Exponents.

The Power of a Quotient Rule states:

• When you divide like bases, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Applying this rule to the given problem:

1. The base of both the numerator and the denominator is $4 \times 7$.
2. The exponent in the numerator is 12, and the exponent in the denominator is 5.
3. Therefore, subtract the exponents: $12 - 5 = 7$.

The simplified expression becomes:

$\left(4\times7\right)^7$.

The solution to the question is: $\left(4\times7\right)^7$

Let's solve the expression step-by-step using the power of a quotient rule for exponents.

The provided expression is:

$\frac{\left(6\times3\right)^7}{\left(3\times6\right)^2}$

The power of a quotient rule states:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

First, simplify the terms inside the parentheses:

• The numerator: $6 \times 3 = 18$
• The denominator: $3 \times 6 = 18$

Thus, the expression simplifies to:

$\frac{18^7}{18^2}$

We can simplify this further using the rule of dividing powers with the same base:

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

Apply the rule:

$\frac{18^7}{18^2} = 18^{7-2} = 18^5$

The solution to the question is:

Let's solve the given expression: $\frac{\left(9\times7\right)^9}{\left(9\times7\right)^4}$.

We need to use the Power of a Quotient Rule for exponents. The rule states that:

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

In our expression, $a = 9 \times 7$, $m = 9$, and $n = 4$.

Applying the rule, we have:

$\frac{\left(9\times7\right)^9}{\left(9\times7\right)^4} = \left(9\times7\right)^{9-4}$

Perform the subtraction in the exponent:

$9 - 4 = 5$

So the expression simplifies to:

$\left(9\times7\right)^5$

Therefore, the expression $\frac{\left(9\times7\right)^9}{\left(9\times7\right)^4}$ simplifies to $\left(9\times7\right)^5$, which is indeed the correct answer.

Let's break down the expression and apply the rules of exponents step by step. We start with the given expression:

$\frac{\left(5\times2\right)^8}{\left(2\times5\right)^{} }$

Step 1: Simplify the denominator

The denominator $\left(2\times5\right)$ is equivalent to $10$, but because it does not have an exponent specified, it is effectively raised to the power of 1:

$\left(2\times5\right)^1 = 10^1 = 10$

Step 2: Apply the power of a quotient rule

The expression $\frac{\left(5\times2\right)^8}{\left(2\times5\right)}$ can be simplified by applying the power of a quotient rule for exponents:

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

Here $a = \left(5 \times 2\right)$, $m = 8$, and the exponent in the denominator is $n = 1$ because$(2 \times 5) = 10^1$. Applying the quotient rule gives:

$\left(5\times2\right)^{8-1}$

This simplifies to:

$\left(5\times2\right)^7$

Final Answer

The simplification results in:

$\left(5\times2\right)^7$

The given expression is:

$\frac{\left(10\times3\right)^{11}}{\left(10\times3\right)^{11}}$

This expression is a fraction where the numerator and the denominator are the same, both equal to $\left(10\times3\right)^{11}$.

According to the quotient rule of exponents, which states that:

• $\frac{a^m}{a^n} = a^{m-n}$, when $a$ is non-zero.

we can simplify the expression by subtracting the exponents in the denominator from the exponent in the numerator.

In our case, applying the formula:

$\frac{\left(10\times3\right)^{11}}{\left(10\times3\right)^{11}} = \left(10\times3\right)^{11-11}$

Which results in:

$\left(10\times3\right)^0$

This simplification uses the rule that any number raised to the power of zero is 1 (as long as the base is not zero). Thus, our final simplified expression $\left(10\times3\right)^0$ is indeed equal to 1.

The solution to the question is: $\left(10\times3\right)^0$.

First, we'll enter the same fraction using the multiplication law between fractions, by multiplying numerator by numerator and denominator by denominator:

$\frac{x}{y}\cdot\frac{w}{z}=\frac{x\cdot w}{y\cdot z}$

Let's return to the problem and apply the above law:

$\frac{a^{12}}{a^9}\cdot\frac{a^3}{a^4}=\frac{a^{12}\cdot a^3}{a^{^9}\cdot a^4}$

From here on we will no longer indicate the multiplication sign, but use the conventional writing method where placing terms next to each other means multiplication.

Now we'll notice that both in the numerator and denominator, multiplication is performed between terms with identical bases, therefore we'll use the power law for multiplication between terms with the same base:

$b^m\cdot b^n=b^{m+n}$

Note that this law can only be used to calculate multiplication between terms with identical bases.

Let's return to the problem and calculate separately the results of multiplication in the numerator and denominator:

$\frac{a^{12}a^3}{a^{^9}a^4}=\frac{a^{12+3}}{a^{9+4}}=\frac{a^{15}}{a^{13}}$

where in the last step we calculated the sum of the exponents.

Now, we'll notice that we need to perform division (fraction=division operation between numerator and denominator) between terms with identical bases, therefore we'll use the power law for division between terms with the same base:

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

Note that this law can only be used to calculate division between terms with identical bases.

Let's return to the problem and apply the above law:

$\frac{a^{15}}{a^{13}}=a^{15-13}=a^2$

where in the last step we calculated the result of subtraction in the exponent.

We got the most simplified expression possible and therefore we're done,

therefore the correct answer is D.