Powers of a Fraction: Variable in the base of the power

To solve this problem, we will apply the power of a fraction rule:

Step 1: Recognize that we are asked to simplify $\left(\frac{x}{y}\right)^8$.

Step 2: Apply the power of a fraction rule, which states:

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

Step 3: Use this formula to obtain:

$\left(\frac{x}{y}\right)^8 = \frac{x^8}{y^8}$

Therefore, the simplified expression of $\left(\frac{x}{y}\right)^8$ is $\frac{x^8}{y^8}$.

The correct choice from the given options is:

$\frac{x^8}{y^8}$

The problem asks us to express $\left(\frac{a}{b}\right)^9$ using exponent rules. We will use the rule for the power of a fraction, which states:

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

Applying this rule, we get:

$\left(\frac{a}{b}\right)^9 = \frac{a^9}{b^9}$

This method ensures that the exponent $9$ is applied to both the numerator and the denominator of the fraction.

Therefore, the solution to the problem is $\frac{a^9}{b^9}$.

We need to rewrite the expression $\left(\frac{a}{3}\right)^2$ using the rule of exponents for fractions. This rule states that if you have a fraction $\left(\frac{m}{n}\right)$ and you raise it to a power $k$, it is equivalent to raising both the numerator and the denominator to the power $k$. Therefore, we have:

$\left(\frac{a}{3}\right)^2 = \frac{a^2}{3^2}$

Here, $a^2$ is the numerator and $3^2$ is the denominator. The expression simplifies to:

$\frac{a^2}{9}$

Based on the provided choices, the correct answer is:

Choice 1: $\frac{a^2}{3^2}$

Therefore, the solution to the given problem is $\frac{a^2}{3^2}$.

To solve this problem, we'll apply the exponent rule for fractions:

• Step 1: Identify the fraction $\frac{b}{5}$ and the power $4$.
• Step 2: Apply the exponent to both the numerator and the denominator, as per the formula.
• Step 3: Use the rule $\left(\frac{b}{5}\right)^4 = \frac{b^4}{5^4}$.

Now, let's work through the application:
Step 1: We have the base fraction $\frac{b}{5}$ and exponent $4$.
Step 2: According to the exponent rule, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, apply the exponent $4$ to both $b$ and $5$.
Step 3: This results in the expression $\frac{b^4}{5^4}$.

Therefore, the expression $\left(\frac{b}{5}\right)^4$ simplifies to $\frac{b^4}{5^4}$.

To solve this problem, we will rewrite the expression $\left(\frac{6}{x}\right)^3$ using the power of a fraction rule. The steps are as follows:

• Identify the fraction's numerator $6$ and denominator $x$.

• According to the power of a fraction rule, apply the power 3 to both the numerator and the denominator:

• $\left(\frac{6}{x}\right)^3 = \frac{6^3}{x^3}$.

Therefore, the expression is correctly written as $\frac{6^3}{x^3}$.

Comparing with the provided answer choices, the correct choice is choice $2$:

$\frac{6^3}{x^3}$

To solve this problem and transform the expression $\left(\frac{5}{y}\right)^7$, we need to utilize the exponent rule for powers of fractions:

• Step 1: Recognize that the expression $\left(\frac{5}{y}\right)^7$ involves both the numerator 5 and the denominator $y$.
• Step 2: According to the exponent rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, we can apply the exponent of 7 to both the numerator and the denominator.
• Step 3: Applying this rule gives us $\frac{5^7}{y^7}$. This step distributes the power to each component of the fraction, preserving the structure of the expression.

Thus, the simplified form of the expression $\left(\frac{5}{y}\right)^7$ is $\frac{5^7}{y^7}$.

This matches choice 3 from the provided options.

To solve the problem, let's apply exponent rules to the given expression:

• Step 1: Identify the fraction's components. The numerator is $4$ and the denominator is $a \times b$.
• Step 2: Apply the exponent rule for fractions, $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$. In this case, $m = 4$, $n = a \times b$, and $p = 2$.

Now, we apply the exponent:

$\left(\frac{4}{a \times b}\right)^2 = \frac{4^2}{(a \times b)^2}$.

This results in:

$\frac{16}{a^2 \times b^2}$.

However, the expression $\frac{4^2}{(a \times b)^2}$ matches choice 2 from the provided options, hence:

The correct answer to the problem in its intended form is $\frac{4^2}{\left(a\times b\right)^2}$.

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

• Step 1: Recognize the expression as an example of a power of a fraction.
• Step 2: Apply the power of a fraction rule to the expression.
• Step 3: Compare the resulting expression with the answer choices.

Now, let's work through each step:
Step 1: The given expression is $\left(\frac{5}{x \times y}\right)^5$.
Step 2: Apply the power of a fraction rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. This gives us:

$\left(\frac{5}{x \times y}\right)^5 = \frac{5^5}{(x \times y)^5}$

Step 3: Compare with answer choices:

• Choice 1: $\frac{5^5}{(x \times y)^5}$ matches our expression exactly.
• Choice 2: $\frac{5^5}{x^5 \times y^5}$ results from distributing the exponent across the product in the denominator, which is another valid interpretation.
• Choice 3: Indicates both expressions in choices 1 and 2 (a'+b') are correct interpretations of the expanded form.
• Choice 4: $\frac{5^5}{x \times y^5}$ is incorrect as it improperly applies the exponent only to $y$.

Therefore, the solution to the problem, which captures both possible expressions, is a'+b' are correct.

To solve this problem, we'll utilize the properties of exponents to simplify the expression $\left(\frac{a \times x}{7}\right)^6$.

Let's proceed with the steps:

• Step 1: Utilize the exponent rule for fractions: $\left( \frac{m}{n} \right)^p = \frac{m^p}{n^p}$. This allows us to express the given expression as:

$\left(\frac{a \times x}{7}\right)^6 = \frac{(a \times x)^6}{7^6}$

• Step 2: Apply the exponent rule to the multinomial in the numerator: $(a \times x)^6 = a^6 \times x^6$.

Thus, we have:

$\frac{(a \times x)^6}{7^6} = \frac{a^6 \times x^6}{7^6}$

Conclusion: The correct expression is $\frac{a^6 \times x^6}{7^6}$, which corresponds to choice 4.

Let's work through the solution step-by-step:

Step 1: Identify the expression
We are given the expression $\left(\frac{a \times b}{x \times y}\right)^2$.

Step 2: Apply the power of a fraction rule
Using the rule $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$, we can rewrite the expression as:

$\frac{(a \times b)^2}{(x \times y)^2}$.

Which matches option 1.

Step 3: Apply the distributive property of exponents over multiplication
Using the rule $(m \times n)^p = m^p \times n^p$, each part of the expression is expanded:

$(a \times b)^2 = a^2 \times b^2$ and $(x \times y)^2 = x^2 \times y^2$.

Thus, the expression becomes:

$\frac{a^2 \times b^2}{x^2 \times y^2}$.

Step 4: Identify the correct choice
Looking at the provided options, choice 3 is:

$\frac{a^2 \times b^2}{x^2 \times y^2}$ and option 1 is$\frac{(a \times b)^2}{(x \times y)^2}$

Both expression matches our derived solution, confirming that the correct answer is choice 4, A+C are correct.

To solve this problem, we'll utilize the rules of exponents, particularly the rule for raising fractions to a power.

• Step 1: Recognize that the expression $\left(\frac{x \times a}{b \times y}\right)^4$ has a fraction raised to a power.
• Step 2: Apply the rule $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$ to distribute the exponent of 4 to both the numerator and the denominator.
• Step 3: This means $\left(\frac{x \times a}{b \times y}\right)^4 = \frac{(x \times a)^4}{(b \times y)^4}$.
• Step 4: Use the power of a product rule: $(m \times n)^p = m^p \times n^p$, which allows us to write $(x \times a)^4$ as $x^4 \times a^4$ and $(b \times y)^4$ as $b^4 \times y^4$.
• Step 5: Substitute these results back into the expression to get $\frac{x^4 \times a^4}{b^4 \times y^4}$.

Therefore, the simplified expression is $\frac{x^4 \times a^4}{\left(b \times y\right)^4}$.

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

• Step 1: Express the given expression $\left(\frac{a \times b}{2 \times x}\right)^3$ using exponent rules for fractions.
• Step 2: Apply the cube power to both the numerator and the denominator separately.
• Step 3: Simplify the expression and compare it to the provided choices.

Now, let's work through each step:

Step 1: We have the expression $\left(\frac{a \times b}{2 \times x}\right)^3$. According to the power of a fraction rule, $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$, we can raise the numerator and denominator to the power of 3 separately.

Step 2: Apply the cube power:

• Numerator: $(a \times b)^3 = a^3 \times b^3$
• Denominator: $(2 \times x)^3 = 2^3 \times x^3 = 8 \times x^3$

Step 3: Combine these results:

The expression simplifies to $\frac{a^3 \times b^3}{8 \times x^3}$.

Comparing it to the given choices, we find that this matches Choice 1: $\frac{a^3 \times b^3}{8 \times x^3}$.

Therefore, the solution to the problem is $\frac{a^3 \times b^3}{8 \times x^3}$.

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

• Step 1: Identify the given expression and its structure.
• Step 2: Apply the rule for powers of a fraction, $\left(\frac{p}{q}\right)^n$.
• Step 3: Calculate powers of individual components within the fraction.
• Step 4: Simplify the resulting expression.

Now, let's work through each step:

Step 1: Our given expression is $\left(\frac{a \times 3}{2 \times x}\right)^3$.

Step 2: Apply the power to the entire fraction using $\left(\frac{p}{q}\right)^n = \frac{p^n}{q^n}$, we get:
$\left(\frac{a \times 3}{2 \times x}\right)^3 = \frac{(a \times 3)^3}{(2 \times x)^3}$.

Step 3: Simplify the numerator and the denominator separately:
Numerator: $(a \times 3)^3 = a^3 \times 3^3 = a^3 \times 27$.
Denominator: $(2 \times x)^3 = 2^3 \times x^3 = 8 \times x^3$.

Step 4: Combine the simplified components to form the final expression:
The expression is $\frac{a^3 \times 27}{8 \times x^3}$.

Therefore, the solution to the problem is $\frac{a^3 \times 27}{8 \times x^3}$, which corresponds to choice 2.

To solve this problem, we'll apply the rule for powers of a fraction:

• Step 1: Identify the fraction. The fraction given is $\frac{6}{x \times y}$.
• Step 2: Apply the power to both the numerator and the denominator. This means squaring both 6 and $x \times y$.
• Step 3: Calculate the square of the numerator and the denominator:
• The square of the numerator: $6^2 = 36$.
• The square of the denominator: $(x \times y)^2 = x^2 \times y^2$.
• Step 4: Combine the results: $\left(\frac{6}{x \times y}\right)^2 = \frac{6^2}{(x \times y)^2} = \frac{36}{x^2 \times y^2}$.

Thus, the expression $\left(\frac{6}{x \times y}\right)^2$ simplifies to $\frac{36}{(x \times y)^2}$.

Therefore, the correct answer is clearly the expression $\frac{36}{(x \times y)^2}$, which matches choice 4.

The task is to simplify $\left(\frac{2\times a}{3}\right)^2$.

First, applying the exponent rule for fractions, $\left(\frac{b}{c}\right)^n = \frac{b^n}{c^n}$, we have:

• $\left(\frac{2\times a}{3}\right)^2 = \frac{(2 \times a)^2}{3^2}$ - which represents performing the exponentiation separately on the numerator and denominator.

Now, simplify each part:

• The numerator: $(2 \times a)^2 = 2^2 \times a^2 = 4 \times a^2$.
• The denominator: $3^2 = 9$.

Thus, the expression simplifies to $\frac{4 \times a^2}{9}$.

We ensure the valid transformations based on the choices provided:

• Choice 1: $\frac{4\times a^2}{9}$, which matches what we calculated.
• Choice 2: $\frac{2^2\times a^2}{3^2}$, which is an equivalent form before final multiplication.
• Choice 3: $\frac{\left(2\times a\right)^2}{3^2}$, presenting the step before breaking down the $(2a)^2$.
• Choice 4: "All answers are correct", recognizing all transformations as valid.

Therefore, each choice represents correct steps or forms towards the simplified expression.

The correct answer is: All answers are correct.

To solve this problem, we'll rewrite the given fraction $\frac{3^8 \times a^8}{x^8 \times 5^8}$ using the rules of powers and exponents.

• Step 1: Recognize the overall structure of the fraction is $\frac{m^8 \times n^8}{p^8 \times q^8}$.
• Step 2: Apply the rule of exponents to express it as a single power of the entire fraction: $\left(\frac{m \times n}{p \times q}\right)^8$.
• Step 3: Substitute the values: $\left(\frac{3 \times a}{x \times 5}\right)^8$ simplifies the expression effectively, given that all components are non-zero.

Thus, by applying the exponent rule directly to the entire fraction, we simplify to $\left(\frac{3 \times a}{x \times 5}\right)^8$.

To solve this problem, our goal is to express the given quotient of powers in a simplified form using exponent laws.

• Step 1: Understand the original expression. We have $\frac{a^5 \times x^5}{7^5 \times b^5}$.
• Step 2: Recognize the structure. Notice that both the numerator and denominator are raised to the fifth power.
• Step 3: Apply the property of exponents for quotients and products, which states that $\left(\frac{m}{n}\right)^k = \frac{m^k}{n^k}$ and $(m \cdot n)^k = m^k \cdot n^k$.
• Step 4: Rewrite the expression as a single fraction raised to the power of 5. Since each term in the numerator and denominator is raised to the fifth power separately, we combine them under a single power:
• Numerator: $a^5 \times x^5 = (a \times x)^5$
• Denominator: $7^5 \times b^5 = (7 \times b)^5$
• Therefore, $\frac{a^5 \times x^5}{7^5 \times b^5} = \left(\frac{a \times x}{7 \times b}\right)^5$.

Thus, the expression can be written as: $\left(\frac{a \times x}{7 \times b}\right)^5$.

Now, comparing this with the answer choices provided:

• Choice 1: $\frac{(a \times x)^5}{7 \times b^5}$ - does not match, as it retains the separate powers incorrectly.
• Choice 2: $\left(\frac{a \times x}{7 \times b}\right)^5$ - matches perfectly as derived.
• Choice 3: $\frac{a \times x^5}{(7 \times b)^5}$ - incorrect form compared to derived structure.
• Choice 4: $7 \times \left(\frac{a \times x}{b}\right)^5$ - unrelated format, doesn't match.

The correct choice is therefore Choice 2. This matches our derived expression using the laws of exponents correctly.

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

• Step 1: Identify the given information: We have the expression $\frac{x^4}{a^4}$.
• Step 2: Apply the appropriate exponent rule: Use the rule $\frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m$.
• Step 3: Simplify the expression using the rule: Substitute $m$ with 4, $x$ with $x$, and $y$ with $a$.

Now, let's work through each step:
Step 1: The expression is $\frac{x^4}{a^4}$.
Step 2: We use the rule $\frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m$, which states that a fraction where the numerator and denominator are raised to the same power can be expressed as a power of a single fraction.
Step 3: Plugging in the values, we have $\frac{x^4}{a^4} = \left(\frac{x}{a}\right)^4$.

Therefore, the solution to the problem is $\left(\frac{x}{a}\right)^4$, which corresponds to choice 1.

To solve this problem, we need to simplify and express the equation $\frac{a^3}{3^3 \times 5^3}$ using exponent rules.

The denominator $3^3 \times 5^3$ can be simplified using the property of exponents: $(ab)^n = a^n \times b^n$. This means that:

$3^3\times5^3=(3\times5)^3$.

Therefore, the expression can be rewritten as:

$\frac{a^3}{(3\times5)^3 }$ which is actually the same as $\left(\frac{a}{3\times5}\right)^3$, using the identity $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$.

Thus, the expression can also be written as:

$\left(\frac{a}{3 \times 5}\right)^3$.

Looking at the provided choices, this expression corresponds to choice marked as $\left(\frac{a}{3 \times 5}\right)^3$.

Therefore, the expression matches both rewritten forms:

The correct answer is a'+b' are correct

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

• Step 1: Recognize the given expression $\frac{4^6}{a^6 \times x^6}$ is a power over a product raised to that power.
• Step 2: Apply the power of a quotient rule to rewrite the expression.

Now, let's work through each step:
Step 1: The given expression is $\frac{4^6}{a^6 \times x^6}$. This can be seen as having common exponents across the numerator and the denominator.
Step 2: Using the power of a quotient rule, which allows us to express the initial expression as $\left(\frac{4}{a \times x}\right)^6$. This step involves recognizing that you can treat the entire $a \times x$ as a single base for the denominator.

Hence, the simplified form of the given expression is $\left(\frac{4}{a \times x}\right)^6$.

Therefore, the solution to the problem is $\left(\frac{4}{a \times x}\right)^6$.