Powers of a Fraction: converting Negative Exponents to Positive Exponents

To solve the problem of converting $\left(\frac{2}{5}\right)^{-2}$ to positive exponents, we use the rule for negative exponents:

Negative exponent rule states:

• $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ — This indicates that we invert the fraction and change the sign of the exponent to positive.

Given expression: $\left(\frac{2}{5}\right)^{-2}$.

Application: By using the rule, the negative exponent instructs us to reciprocate the fraction:

$\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2}$.

The positive exponent $(2)$ indicates the expression is squared. Thus, our action is complete with no further action required.

Thus, the correctly transformed expression of $\left(\frac{2}{5}\right)^{-2}$ is indeed:

$\left(\frac{5}{2}\right)^2$.

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

• Step 1: Identify the given expression with a negative exponent.
• Step 2: Apply the rule for negative exponents, which allows us to convert the expression into a positive exponent form.
• Step 3: Perform the calculation of the new expression.

Now, let's work through each step:

Step 1: The expression given is $\left(\frac{1}{3}\right)^{-4}$, which involves a negative exponent.

Step 2: According to the exponent rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$, we can rewrite the expression with a positive exponent by inverting the fraction:

$\left(\frac{1}{3}\right)^{-4} = \left(\frac{3}{1}\right)^4 = 3^4$.

Step 3: Calculate $3^4$.

The calculation $3^4$ is as follows:

$3^4 = 3 \times 3 \times 3 \times 3 = 81$.

However, since the problem specifically asks for the corresponding expression before calculation to numerical form, the answer remains $3^4$.

Therefore, the answer to the problem, in terms of an equivalent expression, is $3^4$.

To solve the problem, apply the negative exponent rule:

• For any fraction $\frac{a}{b}$ with a negative exponent $-n$, apply the rule: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$.

Apply this rule to the given expression $\left(\frac{10}{13}\right)^{-2}$:

$\left(\frac{10}{13}\right)^{-2} = \left(\frac{13}{10}\right)^{2}$

Therefore, the correct expression with a positive exponent is $\left(\frac{13}{10}\right)^{2}$.

In the provided choices, this is option:

• $\left(\frac{13}{10}\right)^2$

Hence, the correct expression is $\left(\frac{13}{10}\right)^2$.

To solve the problem, we will follow these steps:

• Step 1: Identify the expression and apply the rule for negative exponents.
• Step 2: Take the reciprocal of the given fraction $\frac{10}{17}$.
• Step 3: Raise the reciprocal to the positive power of 5.

Now, let's work through each step:
Step 1: We start with the problem expression $\left(\frac{10}{17}\right)^{-5}$. According to the laws of exponents, a negative exponent means the reciprocal of the base should be raised to the positive of that exponent.
Step 2: Take the reciprocal of $\frac{10}{17}$, which is $\frac{17}{10}$.
Step 3: Raise the reciprocal $\frac{17}{10}$ to the power of 5, resulting in $\left(\frac{17}{10}\right)^5$.

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

To simplify the expression $\left(\frac{1}{20}\right)^{-7}$, we will apply the rule for negative exponents. The key idea is that a negative exponent indicates taking the reciprocal and converting the exponent to a positive:

• Start with the expression: $\left(\frac{1}{20}\right)^{-7}$.
• Apply the negative exponent rule: $\left(\frac{1}{a}\right)^{-n} = a^n$.
• For our expression: $\left(\frac{1}{20}\right)^{-7}$ becomes $20^7$.

Therefore, $\left(\frac{1}{20}\right)^{-7}$ simplifies to $20^7$.

Thus, the correct answer is $20^7$.

To solve the expression $\left(\frac{15}{21}\right)^{-3}$, we will apply the rule for converting negative exponents into positive exponents.

Step 1: Recognize that the negative exponent indicates the reciprocal of the base raised to the positive equivalent of the exponent. Thus, we use the formula:

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

Step 2: Apply this formula to our expression:

$\left(\frac{15}{21}\right)^{-3} = \left(\frac{21}{15}\right)^3$.

Therefore, the solution to the problem is $\left(\frac{21}{15}\right)^3$, which corresponds to choice 3.

To solve for $\left(\frac{1}{60}\right)^{-4}$, we apply the rule for negative exponents.

Step 1: Use the negative exponent rule: For any non-zero number $a$, $a^{-n} = \frac{1}{a^n}$. Thus,

$\left(\frac{1}{60}\right)^{-4} = \left(\frac{60}{1}\right)^4$.

Step 2: Simplify $\left(\frac{60}{1}\right)^4$ by recognizing the identity $\frac{60}{1} = 60$, so it follows that:

$\left(\frac{60}{1}\right)^4 = 60^4$.

Therefore, the simplified expression is $60^4$.

The correct answer is $60^4$

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

• Step 1: Identify the given fraction and the negative exponent.
• Step 2: Apply the conversion rule from negative to positive exponents on fractions.

Now, let's work through each step:
Step 1: The given expression is $\left(\frac{3}{8}\right)^{-5}$. This indicates a fraction raised to a negative power.
Step 2: Applying the rule $(\frac{a}{b})^{-n} = (\frac{b}{a})^{n}$, we invert the fraction and change the exponent to positive. This gives us the expression $\left(\frac{8}{3}\right)^5$.

Therefore, the solution to the problem is $\left(\frac{8}{3}\right)^5$.

To solve this problem, we need to convert the expression $\left(\frac{5}{6}\right)^{-3}$ into a form with positive exponents.

The negative exponent rule states that $x^{-n} = \frac{1}{x^n}$. Applying this to our given fraction:

$\left(\frac{5}{6}\right)^{-3} = \left(\frac{6}{5}\right)^{3}$.

This means we take the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$, and then raise it to the power of 3.

Therefore, the correct expression is $\left(\frac{6}{5}\right)^3$.

This matches choice 1 in the list of possible answers.

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

• Step 1: Recognize that the given expression is $\left(\frac{10 \times 3}{7 \times 9}\right)^{-4}$. A negative exponent indicates that we should take the reciprocal of the base.
• Step 2: Rewrite this expression using the negative exponent rule: $\left(\frac{10 \times 3}{7 \times 9}\right)^{-4} = \left(\frac{7 \times 9}{10 \times 3}\right)^{4}$ This step inverts the fraction and changes the exponent from $-4$ to $4$.
• Step 3: Apply the exponent to each component of the fraction: $\left(\frac{7 \times 9}{10 \times 3}\right)^{4} = \frac{(7 \times 9)^{4}}{(10 \times 3)^{4}}$ This separates the powers between the numerator and the denominator.
• Step 4: Distribute the powers inside each product: $= \frac{7^4 \times 9^4}{10^4 \times 3^4}$ This is achieved by applying $(ab)^n = a^n \times b^n$ to both the numerator and the denominator.

Therefore, the simplified expression is $\frac{7^4 \times 9^4}{10^4 \times 3^4}$, which corresponds to choice 3 in the provided answer choices.

To solve the problem, we need to find the equivalent expression for the given negative exponent:

Step 1: Identify the base fraction as $\frac{6\times8}{2\times7}$.

Step 2: Apply the negative exponent rule:

The expression $\left(\frac{6\times8}{2\times7}\right)^{-5}$ can be rewritten using the property that $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$.

Thus, we have:

$\left(\frac{6\times8}{2\times7}\right)^{-5} = \left(\frac{2\times7}{6\times8}\right)^5$

Therefore, the correct equivalent expression is:

$\left(\frac{2\times7}{6\times8}\right)^5$

Hence, the choice corresponding to this expression is correct.

Therefore, the solution to the problem is $\left(\frac{2\times7}{6\times8}\right)^5$, which corresponds to choice 1.

To solve this problem, let's begin by applying the mathematical rules for negative exponents and exponents of a fraction.

Step 1: Apply the negative exponent rule:

• Given: $\left(\frac{11 \times 9}{4}\right)^{-5}$.

• Using the rule $(a/b)^{-n} = (b/a)^n$, we rewrite this as $\left(\frac{4}{11 \times 9}\right)^5$.

Step 2: Simplify the expression:

• Analyzing the expression $\left(\frac{4}{11 \times 9}\right)^5$, we see that this is equivalent to:

• $\frac{4^5}{(11 \times 9)^5}$.

• Notice that $(11 \times 9)^5$ can also be written as $11^5 \times 9^5$ using properties of exponents.

• Thus, $\frac{4^5}{11^5 \times 9^5}$ is another way to express this fraction.

Step 3: Compare with given choices:

• Choice 2: $\frac{4^5}{11^5 \times 9^5}$ matches our final expression.

• Notice also Choice 3: $\frac{4^5}{(11 \times 9)^5}$ matches the form before simplifying the denominator completely to separate power terms.

Therefore, after comparison, Options B and C are indeed correct and thus the correct response is: B+C are correct.

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

• Step 1: Apply the negative exponent rule.
• Step 2: Use the power of a fraction rule to simplify the expression.
• Step 3: Ensure calculations are conducted correctly, and choose the matching answer from the choices.

Now, let's work through each step:
Step 1: The initial expression is $\left(\frac{3 \times 7}{4 \times 6}\right)^{-6}$. First, use the negative exponent rule: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$. So, the expression becomes $\left(\frac{4 \times 6}{3 \times 7}\right)^{6}$.

Step 2: Apply the power of a fraction rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. Thus, the expression can be re-written as:

$\frac{(4 \times 6)^6}{(3 \times 7)^6}$

Step 3: Assess the provided answer choices and determine which one matches our derived expression. Choice 3, $\frac{(4 \times 6)^6}{(3 \times 7)^6}$, correctly corresponds to our simplified result.

Therefore, the solution to the problem is $\frac{(4 \times 6)^6}{(3 \times 7)^6}$, which corresponds to choice 3.

To simplify the expression $\left(\frac{3\times7}{5\times8}\right)^{-3}$, we follow these steps:

• Step 1: Apply the rule for negative exponents, which states that $a^{-b} = \frac{1}{a^b}$. For fractions, $\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}$.
• Step 2: Rewrite the expression by applying this rule:
  $\left(\frac{3\times7}{5\times8}\right)^{-3} = \frac{(5\times8)^3}{(3\times7)^3}$
• Step 3: Simplify the expression by recognizing the bases to the power of 3: $\frac{(5\times8)^3}{(3\times7)^3}$

The expression can be left as is because it matches one of the choices provided. Performing further calculations here is unnecessary since we have correctly applied the negative exponent rule.

Therefore, the correct simplified form of the expression is $\frac{(5\times8)^3}{(3\times7)^3}$, which corresponds to choice 2.

Let's solve the problem step-by-step:

The given expression is $\left(\frac{4}{9 \times 7}\right)^{-3}$.

Step 1: Apply the negative exponent rule. A negative exponent $-n$ can be transformed by reciprocal and changing the sign of the exponent. Therefore, $\left(\frac{4}{9 \times 7}\right)^{-3} = \left(\frac{9 \times 7}{4}\right)^3$.

Step 2: Express the denominator as a product of integers for clarity: $\frac{9 \times 7}{4}$ is clearer than $\frac{63}{4}$ in context for further steps.

Step 3: Apply the power of a fraction rule. Where $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, therefore, $\left(\frac{9 \times 7}{4}\right)^3 = \frac{(9 \times 7)^3}{4^3}$.

Step 4: Separate the powers within the fraction: The expression $(9 \times 7)^3$ can be expanded using individual exponents: $(9 \times 7)^3 = 9^3 \times 7^3$.

Thus, our expression simplifies to: $\frac{9^3 \times 7^3}{4^3}$.

After analyzing the problem and solving it using the rules of exponents, we conclude that the correct expression is $\frac{9^3 \times 7^3}{4^3}$.

Therefore, the correct choice from the provided options is choice 2: $\frac{9^3 \times 7^3}{4^3}$.

To solve this problem, we will follow the standard rule for evaluating expressions with negative exponents:

• Step 1: Identify the original expression – $\left(\frac{4 \times 5}{3 \times 2}\right)^{-2}$.
• Step 2: Apply the negative exponent rule which states $(a/b)^{-n} = (b/a)^{n}$.
• Step 3: Simplify the expression.

Now, let's proceed through each step:

Step 1: The expression within the parentheses is $\frac{4\times5}{3\times2}$, which simplifies to $\frac{20}{6}$ or further reduced to $\frac{10}{3}$. However, for the purpose of matching the choices given, we'll use the product form directly.

Step 2: Apply the negative exponent formula:
$\left(\frac{4 \times 5}{3 \times 2}\right)^{-2} = \left(\frac{3 \times 2}{4 \times 5}\right)^2$

Step 3: The result of this step shows the reciprocal raised to the power of 2:

The solution to the problem is therefore represented as:
$\left(\frac{3 \times 2}{4 \times 5}\right)^2$.

Upon reviewing the choices provided, this corresponds to choice 1.

To solve the problem of evaluating $(\frac{7}{8})^{-2}$, we'll proceed with these steps:

• Step 1: Convert the negative exponent into a positive exponent by taking the reciprocal of the base.
• Step 2: Evaluate the expression obtained after the conversion.

Now, let's work through each step:

Step 1: Convert the negative exponent to a positive exponent using the reciprocal:
Using the property $(\frac{a}{b})^{-n} = (\frac{b}{a})^{n}$, we have:

$(\frac{7}{8})^{-2} = (\frac{8}{7})^{2}$

Step 2: Calculate the positive power:

$(\frac{8}{7})^{2} = \frac{8^2}{7^2} = \frac{64}{49}$

Thus, the solution to the problem is:
$(\frac{7}{8})^{-2} = \frac{64}{49}$

Extra step to express $\frac{64}{49}$ as a mixed number:

$64 \div 49 = 1$ remainder $15$, so $\frac{64}{49} = 1 \frac{15}{49}$.

Therefore, the simplified solution to the expression $(\frac{7}{8})^{-2}$ is $1 \frac{15}{49}$.

To solve this problem, we'll apply the rules for negative exponents:

We start with the expression: $\left(\frac{7 \times 11 \times 19}{3 \times 12 \times 15}\right)^{-6}$.

By the rule for negative exponents, $(a/b)^{-n} = (b/a)^n$, we invert the fraction and change the exponent from $-6$ to $6$:

$\left(\frac{3 \times 12 \times 15}{7 \times 11 \times 19}\right)^6$

This transformation capitalizes on the idea that negative exponents reflect a reciprocal relationship.

Therefore, the expression with positive exponents is: $\left(\frac{3 \times 12 \times 15}{7 \times 11 \times 19}\right)^6$.

We begin with the expression: $\frac{1}{a^{-x}}$.
Our goal is to simplify this expression while converting any negative exponents into positive ones.

• Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$.
• Correspondingly, $\frac{1}{a^{-n}} = a^n$.
• Thus, in our expression $\frac{1}{a^{-x}}$, the negative exponent can be converted and flipped to the numerator by the rule: $\frac{1}{a^{-x}} = a^x$.

We are given the expression: $\left(\frac{1}{2\times3\times4}\right)^{-2}$. We need to simplify it using the rules of exponents.

• Step 1: Identify the base of the exponent.
  The base is $\frac{1}{2\times3\times4}$.
  

• Step 2: Apply the rule for negative exponents.
  For a fraction $\frac{1}{a}$ with a negative exponent, $\left( \frac{1}{a} \right)^{-n} = a^n$. Therefore, $\left(\frac{1}{2\times3\times4}\right)^{-2} = (2\times3\times4)^2$.
  

• Step 3: Expand the expression.
  $(2\times3\times4)^2 = 2^2 \times 3^2 \times 4^2$.

Thus, the simplified expression is: $2^2\times3^2\times4^2$