Powers of a Fraction: Inverse formula

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 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 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, 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, 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, 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'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 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 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, 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, 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 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.

Let's solve the problem step by step:

We start with the expression:

$\frac{7^2 \times x^2}{a^2}$.

Recognizing the terms in the expression, we notice that:

• $7^2$ is the square of $7$.
• $x^2$ is the square of $x$.
• $a^2$ is the square of $a$.

Using one of the properties of exponents, we know that:

$\frac{b^m \times c^m}{d^m} = \left(\frac{b \times c}{d}\right)^m$.

Thus, we can rewrite our given expression:

$\frac{7^2 \times x^2}{a^2}$ can be rewritten as $\left(\frac{7 \times x}{a}\right)^2$.

This conversion works because the squares in the numerator and the denominator allow us to apply the rule of powers over fractions.

The equivalent expression is therefore $\left(\frac{7 \times x}{a}\right)^2$.

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 simplify the given expression $\frac{6^8 \times 7^8}{17^8}$, we'll use the following steps:

• Step 1: Apply the power of a product rule. We know that $a^n \times b^n = (a \times b)^n$.
• Step 2: Simplify the expression by recognizing that the numerator $6^8 \times 7^8$ can be rewritten using the power of a product rule.

Now, let's simplify the numerator:

$6^8 \times 7^8 = (6 \times 7)^8$

Thus, the expression becomes:

$\frac{(6 \times 7)^8}{17^8}$

This matches the form given in choice $\frac{\left(6\times7\right)^8}{17^8}$. Therefore, this is the correct simplification of the original expression.

Therefore, the correct answer is $\frac{(6 \times 7)^8}{17^8}$, corresponding to choice 3.

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 aim to rewrite the given expression using the properties of exponents. The expression we need to deal with is $\frac{6^5}{13^5 \times 4^5}$.

We can simplify this using the formula for powers of fractions, which states that if two exponents are the same, we can treat the fraction as a whole raised to that exponent: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$.

Applying this to the problem, considering the expression $\frac{6^5}{(13 \times 4)^5}$ all raised to the power of 5 can be rewritten, using our formula, as a single fraction raised to the same power:
$\left(\frac{6}{13 \times 4}\right)^5$.

This simplifies the entire expression to a clear fraction raised to the power 5. Therefore, the corresponding expression to the original problem is:

$\left(\frac{6}{13 \times 4}\right)^5$.