Power of a Quotient Rule for Exponents: Variable in the base of the power

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

• Step 1: Identify the common base in both the numerator and the denominator.

• Step 2: Apply the Power of a Quotient Rule for Exponents.

• Step 3: Simplify the expression by subtracting the exponents.

Now, let's work through each step:
Step 1: Notice that both the numerator $x^6$ and the denominator $x^2$ share the same base, $x$.
Step 2: The Power of a Quotient Rule states that $\frac{a^m}{a^n} = a^{m-n}$. We apply this rule to our expression, obtaining $x^{6-2}$.
Step 3: Simplifying $6 - 2$, we find that the expression simplifies to $x^4$.

Therefore, the simplified form of the expression $\frac{x^6}{x^2}$ is $x^4$.

Considering the given answer choices:

• Choice 1: $x^{6+2}$ is incorrect because it involves adding the exponents, which does not follow the rules for division of powers.

• Choice 2: $x^{6-2}$ is the correct as the setup simplification, and can be fully simplified to yield $x^4$ for clarity.

• Choice 3: $x^{6\times2}$ is incorrect because it multiplies the exponents, which is not applicable in division.

• Choice 4: $x^{\frac{6}{2}}$ is not directly applicable as it assumes a different interpretation not aligning with subtraction of exponents for division.

The correct choice is represented by choice 2, $x^{6-2}$.

To solve this problem, we'll follow a systematic approach to simplify $\frac{a^9}{a^4}$ using exponent rules:

• Step 1: Identify the expression.
  We are given $\frac{a^9}{a^4}$ and asked to simplify it.
• Step 2: Apply the Quotient Rule for Exponents.
  According to the quotient rule, when dividing two expressions with the same base, we subtract the exponents. Given
  $\frac{a^m}{a^n} = a^{m-n}$,
  we apply this rule to our expression:

$a^{9-4}$

• Step 3: Simplify the Expression.
  We simplify the exponent by performing the subtraction:
  $a^{9-4} = a^5$.

Thus, the simplified form of the expression $\frac{a^9}{a^4}$ is $\mathbf{a^5}$.

Now, let's match our simplified expression with the provided choices:

• Choice 1: $a^{9-4}$.
  This reflects the correct expression before computation.
• Choice 2: $a^{9+4}$.
  Incorrect, as it adds the exponents instead of subtracting.
• Choice 3: $a^{9\times4}$.
  Incorrect, as it multiplies the exponents instead of subtracting.
• Choice 4: $a^{\frac{9}{4}}$.
  Incorrect, as it divides the exponents instead of subtracting.

Hence, the correct choice is Choice 1: $a^{9-4}$, which is the correct representation before simplification.

I am confident in the correctness of this solution.

To solve this problem, we will apply the Quotient Rule for exponents, which helps simplify expressions where both the numerator and denominator share the same base.

• Step 1: Identify the given information.
  We are given the expression $\frac{b^{11}}{b^8}$.
• Step 2: Apply the Quotient Rule for exponents.
  According to the Quotient Rule, $\frac{b^m}{b^n} = b^{m-n}$. Here, $m = 11$ and $n = 8$.
• Step 3: Subtract the exponent of the denominator from the exponent of the numerator.
  Calculate $11 - 8 = 3$, leading to the simplified expression $b^3$.

Therefore, the simplified form of the given expression is $b^{3}$.

When considering the choices:

• Choice 1: $b^{11+8}$ adds the exponents, which is incorrect for division.
• Choice 2: $b^{11\times8}$ multiplies the exponents, also incorrect for division.
• Choice 3: $b^{\frac{11}{8}}$ divides the exponents, which is incorrect for division.
• Choice 4: $b^{11-8}$, correctly subtracts the exponents as per the Quotient Rule.

Thus, the correct choice is Choice 4: $b^{11-8}$, which simplifies to $b^3$.

We are given the expression: $\frac{y^{20}}{y^{11}}$

To solve this, we will use the rule for exponents known as the Power of a Quotient Rule, which states that when you divide two expressions with the same base, you can subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Applying this rule to our expression:

$\frac{y^{20}}{y^{11}} = y^{20-11}$

After simplifying, we have:

$y^{9}$

The solution to the question is: $y^{9}$

We are given the expression: $\frac{x^{18}}{x^7}$.

To simplify this, we use the Power of a Quotient Rule for Exponents. This rule states that when dividing like bases, you subtract the exponent in the denominator from the exponent in the numerator.

So, according to this rule:
$\frac{x^m}{x^n} = x^{m-n}$.

Apply this rule to our expression: $\frac{x^{18}}{x^7} = x^{18-7}$.

Simplify the exponent by subtracting: $18-7 = 11$.

Therefore, the simplified expression is: $x^{11}$.

However, the expected form of the answer once applying the rule (before simplification) is: $x^{18-7}$.

The solution to the question is: $x^{18-7}$.

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: $x^7$

Since a division operation between two terms with identical bases is required, we will use the power property to divide terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$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 mentioned power property:

$\frac{a^a}{a^b}=a^{a-b}$Therefore, the correct answer is option D.

We are tasked with simplifying the expression $\frac{\left(x \times y\right)^{25}}{\left(x \times y\right)^{21}}$.

To solve this, we will apply the Power of a Quotient Rule for Exponents. This rule states that when you divide powers with the same base, you subtract the exponents. Formally, $\frac{a^m}{a^n} = a^{m-n}$.

Here, the base $a$ is $(x \times y)$, and the exponents $m$ and $n$ are 25 and 21, respectively. So, applying this rule gives us:

$\frac{\left(x \times y\right)^{25}}{\left(x \times y\right)^{21}} = \left(x \times y\right)^{25 - 21}$

By simplifying the expression $25 - 21$, we get:

$\left(x \times y\right)^{4}$

Thus, the solution to the problem is effectively represented by the expression $\left(x \times y\right)^{25 - 21}$.

Now, looking at the choices provided:

• Choice 1: $\left(x \times y\right)^{\frac{25}{21}}$ is incorrect, as it uses division instead of subtraction.
• Choice 2: $\left(x \times y\right)^{25-21}$ is correct, as it applies the Quotient Rule correctly.
• Choice 3: $\left(x \times y\right)^{25 \times 21}$ is incorrect because it uses multiplication instead of subtraction.
• Choice 4: $\left(x \times y\right)^{25+21}$ is incorrect because it uses addition instead of subtraction.

Therefore, the correct answer is choice 2, $\left(x \times y\right)^{25-21}$.

I am confident in the correctness of this solution based on the consistent application of mathematical rules.

The given expression is: $\frac{\left(x\times a\right)^{30}}{\left(a\times x\right)^{15}}$

To solve this, we can apply the quotient rule for exponents. The quotient rule states that $\frac{b^m}{b^n} = b^{m-n}$, where $b$ is the base and $m$ and $n$ are the exponents.

In this problem, both the numerator and the denominator have the same base $\left(x \times a\right)$. Thus, the expression simplifies by subtracting the exponents:

• The exponent of the numerator is 30.
• The exponent of the denominator is 15.

Applying the power of a quotient rule, we have:

$\left(x \times a\right)^{30-15}$

Thus, the simplified expression is $\left(x \times a\right)^{15}$.

The solution to the question is: $\left(x \times a\right)^{15}$.

To solve the given expression, we apply the Power of a Quotient Rule for Exponents. This rule tells us that if we have an expression of the form $\frac{b^m}{b^n}$, it simplifies to $b^{m-n}$.

Given the expression $\frac{(2\times a)^5}{(2\times a)^3}$, we can identify it with the rule as follows. Here, the base $(2\times a)$ is the same in both the numerator and the denominator, with exponents 5 and 3 respectively.

According to the rule, we subtract the exponent in the denominator from the exponent in the numerator, which results in $(2\times a)^{5-3}$.

This simplifies to $(2\times a)^2$, but based on the way the answer is expected to be expressed, we stick with $(2\times a)^{5-3}$.

Thus, the solution to the question is: $(2\times a)^{5-3}$

Let's solve the expression $\frac{(3 \times x)^8}{(3 \times x)}$ step by step using the Power of a Quotient Rule for Exponents.

The expression given is:

$\frac{(3 \times x)^8}{(3 \times x)}$

The Power of a Quotient Rule states that for any non-zero number $a$, and integers $m$ and $n$, the expression $\frac{a^m}{a^n}$ is equal to $a^{m-n}$.

In this problem, $a$ is $3 \times x$, $m = 8$, and $n = 1$.

Applying the Power of a Quotient Rule:

• Subtract the exponent in the denominator from the exponent in the numerator. So we have $(3 \times x)^{8-1}$.

Thus, the simplified form of the expression is:

$(3 \times x)^{7}$

The solution to the question is: $(3 \times x)^{7}$.

To solve the problem $\frac{\left(a \times b\right)^{12}}{\left(a \times b\right)^3}$, we can use the rule for exponents known as the Power of a Quotient Rule, which states that $\frac{x^m}{x^n} = x^{m-n}$, where $x$ is a non-zero base and $m$ and $n$ are the exponents.

Let's apply this rule step by step to our expression:

• Identify the base: In the expression $\frac{\left(a \times b\right)^{12}}{\left(a \times b\right)^3}$, the base is $a \times b$.
• Identify the exponents: The exponent for the numerator is 12, and for the denominator, it is 3.
• Apply the Power of a Quotient Rule: $\frac{\left(a \times b\right)^{12}}{\left(a \times b\right)^3} = \left(a \times b\right)^{12-3}$.

Thus, the simplification of the given expression is: $\left(a \times b\right)^{12-3}$

The solution to the question is: $\left(a \times b\right)^{9}$

To solve the expression $\frac{(7\times x)^9}{(7\times x)}$, we can utilize the Power of a Quotient Rule for exponents. According to this rule, for any non-zero number $a$ and integers $m$ and $n$, the expression $\frac{a^m}{a^n}$ can be simplified to $a^{m-n}$.

Applying this rule, we identify the base $(7\times x)$ as the variable and analyze the exponents:

• The numerator is $(7\times x)^9$, which means the power 9 applies to the term $(7\times x)$.
• The denominator is $(7\times x)^1$, which implies a power of 1.

Now, we apply the quotient rule:

$\frac{(7\times x)^9}{(7\times x)} = (7\times x)^{9-1} = (7\times x)^8$

Thus, the expression simplifies to $(7\times x)^8$. This is achieved by subtracting the exponent in the denominator from the exponent in the numerator.

The solution to the question is: $(7\times x)^8$.

The given expression is:
$\frac{\left(a\times b\right)^{15}}{\left(a\times b\right)^3}$

We need to apply the division rule for exponents, which states that:
$\frac{x^m}{x^n} = x^{m-n}$

Using this rule, we can simplify the expression by subtracting the exponent of the denominator from the exponent of the numerator:
$\frac{\left(a\times b\right)^{15}}{\left(a\times b\right)^3} = \left(a\times b\right)^{15-3}$

Subtracting the exponents, we have:
$\left(a\times b\right)^{12}$

Therefore, the simplified expression is:
$\left(a\times b\right)^{12}$

The solution to the question is:
$\left(a\times b\right)^{12}$

To solve the problem, we need to simplify the expression $\frac{\left(x\times y\right)^{27}}{\left(x\times y\right)^{20}}$.

We will apply the Power of a Quotient Rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$.

Let's denote $a = x \times y$, and our expression becomes $\frac{a^{27}}{a^{20}}$. According to the rule:

• $m = 27$
  
• $n = 20$
  

We subtract the exponents: $m - n = 27 - 20 = 7$.

Thus, $a^{m-n} = a^7 = (x \times y)^7$.

Therefore, the expression simplifies to $\left(x \times y\right)^7$.

The solution to the question is:

$\left(x\times y\right)^7$

We start with the given expression:
$\frac{\left(a\times x\right)^{21}}{\left(x\times a\right)^6}$.

Firstly, notice that the terms inside the parentheses in both the numerator and the denominator are the same: $a \times x$ and $x \times a$, which are equivalent due to the commutative property of multiplication.

Thus, we can rewrite the expression in terms of $(ax)$ as follows:
$\frac{\left(a x\right)^{21}}{\left(a x\right)^6}$.

We will apply the power of a quotient rule for exponents, which states that:

• $\frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$.

Using this rule to simplify the expression, we have:

$\frac{(ax)^{21}}{(ax)^6} = (ax)^{21-6} = (ax)^{15}$.

Thus, the expression simplifies to $(a \times x)^{15}$.

Therefore, the solution to the question is:
$\left(x\times a\right)^{15}$.