Powers of a Fraction: Inverse formula

To solve this problem, we'll use the rule of negative exponents:

• Step 1: Identify that the given expression is $\frac{1}{3^2}$.
• Step 2: Recognize that $\frac{1}{3^2}$ can be rewritten using the negative exponent rule.
• Step 3: Apply the formula $\frac{1}{a^n} = a^{-n}$ to the expression $\frac{1}{3^2}$.

Now, let's work through these steps:

Step 1: We have $\frac{1}{3^2}$ where 3 is the base and 2 is the exponent.

Step 2: Using the formula, convert the denominator $3^2$ to $3^{-2}$.

Step 3: Thus, $\frac{1}{3^2} = 3^{-2}$.

Therefore, the solution to the problem is $3^{-2}$.

To solve the given problem, we need to express $\frac{1}{5^2}$ using negative exponents. We'll apply the formula for negative exponents, which is $\frac{1}{a^n} = a^{-n}$:

• Identify the base and power in the denominator. Here, the base is $5$ and the power is $2$.
• Apply the inverse formula: $\frac{1}{5^2} = 5^{-2}$.

Thus, the equivalent expression for $\frac{1}{5^2}$ using a negative exponent is $5^{-2}$.

To solve this problem, we'll apply the formula for the power of a quotient:

• Step 1: Identify the expression $\frac{1^7}{9^7}$.
• Step 2: Recognize that $\frac{a^n}{b^n} = \left( \frac{a}{b} \right)^n$ applies here. The numerator is $a^7 = 1^7$, and the denominator is $b^7 = 9^7$.
• Step 3: Apply the formula: $\frac{1^7}{9^7} = \left( \frac{1}{9} \right)^7$.

In step 2, we used the property that allows us to rewrite $\frac{1^7}{9^7}$ as $\left( \frac{1}{9} \right)^7$, which is more convenient for interpretation or further calculations.

Therefore, the expression $\frac{1^7}{9^7}$ can be rewritten as $\left( \frac{1}{9} \right)^7$.

To solve this problem, we'll employ the exponent rules for fractions:

• Step 1: Recognize that $\frac{2^9}{11^9}$ follows the general form $\frac{a^n}{b^n}$.
• Step 2: Apply the property $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.
• Step 3: Substitute into the property to express the fraction as $\left(\frac{2}{11}\right)^9$.

Let's work through the steps in detail:

Step 1: The expression $\frac{2^9}{11^9}$ can be viewed as each number, 2 and 11, raised to the 9th power in a fraction.

Step 2: Utilize the exponent rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to rewrite the fraction with a single power.

Step 3: Therefore, the expression $\frac{2^9}{11^9}$ simplifies to $\left(\frac{2}{11}\right)^9$.

Therefore, the correct answer is indeed $\left(\frac{2}{11}\right)^9$.

The correct choice from the provided options is:

$\left(\frac{2}{11}\right)^9$

To solve this problem, we need to express $\frac{1^5}{6^5}$ using the power of a fraction rule:

• Step 1: Identify that both the numerator and denominator are raised to the same power, 5.
• Step 2: Recognize that the expression can be rewritten as $\left(\frac{1}{6}\right)^5$ using the power of a fraction rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

Applying the formula, we convert $\frac{1^5}{6^5}$ into $\left(\frac{1}{6}\right)^5$.

Therefore, the solution to the problem and correct multiple-choice answer is $\left(\frac{1}{6}\right)^5$, which corresponds to choice 2.

To solve the problem of expressing $\frac{1}{4^2}$ using powers with negative exponents:

• Identify the base in the denominator: 4 raised to the power 2.
• Apply the rule for negative exponents that states $\frac{1}{a^n} = a^{-n}$.
• Express $\frac{1}{4^2}$ as $4^{-2}$.

Thus, the expression $\frac{1}{4^2}$ can be rewritten as $4^{-2}$.

To solve this problem, we recognize the application of exponent rules for powers of fractions. The expression $\frac{12^3}{23^3}$ can be rewritten by using the formula $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$.

Let's apply this to the given problem:

• Step 1: Identify the structure as $\frac{a^m}{b^m}$, where $a = 12$, $b = 23$, and $m = 3$.
• Step 2: Use the formula $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ to transform $\frac{12^3}{23^3}$ into $\left(\frac{12}{23}\right)^3$.

The expression $\frac{12^3}{23^3}$ simplifies to $\left(\frac{12}{23}\right)^3$.

Therefore, the correct corresponding expression is $\left(\frac{12}{23}\right)^3$.

To solve this problem, we will apply the exponent rule for powers of a fraction.

• Step 1: Understand the given expression $\frac{2^4}{7^4}$.
• Step 2: Use the formula $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ to rewrite the expression.
• Step 3: Apply this rule to rewrite $\frac{2^4}{7^4} = \left(\frac{2}{7}\right)^4$.

This shows that instead of writing separate powers for the numerator and denominator, we can express it as a single fraction raised to that power.

Thus, the expression $\frac{2^4}{7^4}$ corresponds to $\left(\frac{2}{7}\right)^4$.

The correct choice from the given options is:

• Choice 3: $\left(\frac{2}{7}\right)^4$

Therefore, the solution to the problem is $\left(\frac{2}{7}\right)^4$.

To solve this problem, let's follow these steps:

• Step 1: Identify the given expression. We have $\frac{1^2}{3^2}$.
• Step 2: Apply the appropriate rule for powers of fractions: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$.
• Step 3: Simplify the expression using this rule.

Now, let's proceed through each step in detail:

Step 1: We start with the given expression $\frac{1^2}{3^2}$.

Step 2: According to the rule for powers of fractions, we write this expression as:
$\frac{1^2}{3^2} = \left(\frac{1}{3}\right)^2$.

Step 3: This simplification converts both the numerator and the denominator's power into a single power of the fraction $\left(\frac{1}{3}\right)$.

Therefore, the expression $\frac{1^2}{3^2}$ is equivalent to $\left(\frac{1}{3}\right)^2$.

Comparing with the given answer choices, the correct choice is \( \text{Choice 2: } $\left(\frac{1}{3}\right)^2$.

To solve this problem, we will rewrite the expression $\frac{1}{6^7}$ using the rules of exponents:

Step 1: Identify the given fraction.

We start with $\frac{1}{6^7}$, where the base in the denominator is 6, and the exponent is 7.

Step 2: Apply the formula for negative exponents.

The formula $a^{-n} = \frac{1}{a^n}$ allows us to rewrite a reciprocal power as a negative exponent. This means the expression $\frac{1}{6^7}$ can be rewritten as $6^{-7}$.

Step 3: Conclude with the answer.

By transforming $\frac{1}{6^7}$ to its equivalent form using negative exponents, the expression becomes $6^{-7}$.

Therefore, the correct expression is $\boxed{6^{-7}}$, which corresponds to choice 2 in the given options.

To solve this problem, we will use the properties of exponents. Specifically, we will convert the expression $\frac{1}{20^2}$ into a form that uses a negative exponent. The general relationship is that $\frac{1}{a^n} = a^{-n}$.

Applying this rule to the given expression:

• Step 1: Identify the current form, which is $\frac{1}{20^2}$.
• Step 2: Apply the negative exponent rule: $\frac{1}{20^2} = 20^{-2}$.
• Step 3: This expression, $20^{-2}$, represents $\frac{1}{20^2}$ using a negative exponent.

Therefore, the expression $\frac{1}{20^2}$ can be expressed as $20^{-2}$, which aligns with choice 1.

To solve this problem, we will apply the rule of exponentiation for fractions. This rule states that $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$, where $a$ and $b$ are non-zero numbers and $n$ is an integer.

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

• Step 1: Recognize that the expression we need to rewrite is $\frac{3^6}{8^6}$.
• Step 2: Apply the power of a fraction rule. According to this rule, $\frac{3^6}{8^6} = \left(\frac{3}{8}\right)^6$.
• Step 3: Thus, the expression $\frac{3^6}{8^6}$ simplifies to $\left(\frac{3}{8}\right)^6$.

The solution to the problem is that the expression can be rewritten as $\left(\frac{3}{8}\right)^6$.

To solve the given problem, we want to rewrite the expression $\frac{10^5}{17^5}$ using the rules of exponents.

• Step 1: Recognize that both the numerator and the denominator are raised to the 5th power.
• Step 2: Apply the rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, which allows us to combine the power into a single expression.

By applying this rule, we have:

$\frac{10^5}{17^5} = \left(\frac{10}{17}\right)^5$

This shows that the original expression can be rewritten as a single power of a fraction.

Therefore, the simplified form of the expression is $\left(\frac{10}{17}\right)^5$.

To solve this problem, we need to rewrite the given expression $\frac{20^4}{31^4}$ using properties of exponents.

Let's take these steps:

• Step 1: Recognize the expression as a fraction raised to a power. The problem provides $\frac{20^4}{31^4}$.
• Step 2: Apply the power of a fraction rule: For any real numbers $a$ and $b$, and a positive integer $n$, $\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$.

Applying Step 2, we write:

$\frac{20^4}{31^4} = \left(\frac{20}{31}\right)^4$.

Thus, the corresponding expression is $\left(\frac{20}{31}\right)^4$.

Therefore, the solution to the problem is $\left(\frac{20}{31}\right)^4$.

To solve this problem, let's transform the expression $\frac{7^{10}}{9^{10}}$.

• Step 1: Identify the Form

The expression $\frac{7^{10}}{9^{10}}$ fits the pattern $\frac{a^m}{b^m}$.

• Step 2: Apply the Power of a Quotient Rule

The power of a quotient formula is $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$.

Substitute $a = 7$, $b = 9$, and $m = 10$ into this formula, and we have:

$\frac{7^{10}}{9^{10}} = \left(\frac{7}{9}\right)^{10}$.

We can see that this transformation results in the expression $\left(\frac{7}{9}\right)^{10}$, which matches answer choice 1.

Therefore, the final expression is $\left(\frac{7}{9}\right)^{10}$.

Thus, the correct reformulated expression is $\left(\frac{7}{9}\right)^{10}$.

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: Apply the product of powers formula

• Step 2: Simplify the combined powers into a single fraction with one exponent

Now, let's work through each step:
Step 1: Using $(a^m \times b^m) = (a \times b)^m$, we get:
$(6^6 \times 11^6) = (6 \times 11)^6$ and $(5^6 \times 13^6) = (5 \times 13)^6$.

Step 2: Simplifying the expression, we get:
$\frac{(6^6 \times 11^6)}{(5^6 \times 13^6)} = \frac{(6 \times 11)^6}{(5 \times 13)^6} = \left(\frac{6 \times 11}{5 \times 13}\right)^6$.

This transformation matches both option B and C in the provided answer choices.

Therefore, the correct answer is$\textbf{B + C are correct}$.

To simplify the expression $\frac{5^{10}\times8^{10}}{4^{10}\times7^{10}}$, we start by applying the property of exponents: $(a \times b)^n = a^n \times b^n$.

Step 1: Rewrite the expression. Notice that both the numerator and the denominator consist of two numbers, each raised to the power of 10.

$\frac{5^{10} \times 8^{10}}{4^{10} \times 7^{10}} = \frac{(5 \times 8)^{10}}{(4 \times 7)^{10}}$

Now, we notice we can apply the equality for exponential simplification:

$\left(\frac{5 \times 8}{4 \times 7}\right)^{10} = \frac{(5 \times 8)^{10}}{(4 \times 7)^{10}}$

Concluding, the simplified expression of the given problem is equivalent to option "1":

The expression simplifies to $\frac{\left(5\times8\right)^{10}}{\left(4\times7\right)^{10}}$, which aligns perfectly with choice id="1".

Therefore, the final answer is:

$\frac{\left(5\times8\right)^{10}}{\left(4\times7\right)^{10}}$.

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, 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 $\left(\frac{4}{a \times x}\right)^6$.