Powers of a Fraction: Calculating powers with negative exponents

To solve the expression $\left(\frac{10}{13}\right)^{-4}$, we start by applying the rule for dividing exponents is:

$\frac{10^{-4}}{13^{-4}}$, which maintains the negative exponent but as separate components of fraction resulting in the same value.

Consequently, the expression $\left(\frac{10}{13}\right)^{-4}$ equates to $\frac{10^{-4}}{13^{-4}}$.

By comparing this with the presented choices, we identify that option (2):

$\frac{10^{-4}}{13^{-4}}$

matches correctly with our conversion of the original expression.

Therefore, the correct expression is $\frac{10^{-4}}{13^{-4}}$.

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

We use the formula:

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

Therefore, we obtain:

$(\frac{3}{2})^4$

We use the formula:

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

Therefore, we obtain:

$\frac{3^4}{2^4}=\frac{3\times3\times3\times3}{2\times2\times2\times2}=\frac{81}{16}$

The expression we are given is $\left(\frac{3\times7}{5\times8}\right)^{-3}$. In order to simplify it, we will apply the rules for negative exponents and powers of a fraction.

Step 1: Recognize that we are dealing with a negative exponent. The rule for negative exponents is $a^{-n} = \frac{1}{a^n}$. Thus, we invert the fraction and change the sign of the exponent:

$\left(\frac{3 \times 7}{5 \times 8}\right)^{-3} = \left(\frac{5 \times 8}{3 \times 7}\right)^{3}$

Step 2: Apply the power of a fraction rule, which states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$:

$\left(\frac{5 \times 8}{3 \times 7}\right)^{3} = \frac{(5 \times 8)^3}{(3 \times 7)^3}$

Step 3: Apply the power of a product rule, which allows us to distribute the exponent across the multiplication:

$\frac{(5)^3 \times (8)^3}{(3)^3 \times (7)^3} = \frac{5^3 \times 8^3}{3^3 \times 7^3}$

Step 4: Express each base raised to the power of -3 directly:

$\frac{5^{-3} \times 8^{-3}}{3^{-3} \times 7^{-3}}$

Since the inverted version of the expression can also mean distributing -3 directly across the original fraction components, this can be rearranged as:

$\frac{3^{-3} \times 7^{-3}}{5^{-3} \times 8^{-3}}$

Comparing with the given choices, the corresponding expression is choice 3:

$\frac{3^{-3}\times7^{-3}}{5^{-3}\times8^{-3}}$

Therefore, the equivalent expression for the given problem is $\frac{3^{-3}\times7^{-3}}{5^{-3}\times8^{-3}}$.

To solve this problem, we need to simplify the expression $\left(\frac{10 \times 3}{3 \times 9}\right)^{-4}$.

• Step 1: Simplify the expression inside the parentheses: $\frac{10 \times 3}{3 \times 9} = \frac{30}{27} = \frac{10}{9}$.
• Step 2: We now have $\left(\frac{10}{9}\right)^{-4}$.
• Step 3: Apply the rule for negative exponents: $\left(\frac{10}{9}\right)^{-4} = \left(\frac{9}{10}\right)^4$.
• Step 4: Apply the power of a quotient rule: $\left(\frac{9}{10}\right)^4 = \frac{9^4}{10^4}$.
• Step 5: Expand using exponentiation properties: $\frac{9^4}{10^4} = \frac{(3^2)^4}{10^4} = \frac{3^8}{10^4}$.
• Step 6: Comparing with the given choices, apply rules uniformly: distribute $4$ across numerators and denominators of the specific expression directly.
• Step 7: Write using property: $\left(\frac{10 \times 3}{3 \times 9}\right)^{-4} = \frac{10^{-4}\times3^{-4}}{3^{-4}\times9^{-4}}$.

Therefore, the correct simplified expression is $\frac{10^{-4}\times3^{-4}}{3^{-4}\times9^{-4}}$.

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

• Step 1: Simplify the original expression inside the parentheses.
• Step 2: Apply the negative exponent to each factor in the fraction.
• Step 3: Use exponent rules to rewrite the expression.

Now, let's work through each step:
Step 1: The original expression is $\left(\frac{4 \times 6}{5 \times 10}\right)^{-6}$. Simplifying inside the fraction gives us $\frac{24}{50}$. However, we will work with the original form for clarity.
Step 2: Apply the exponent of $-6$ to each factor separately. This means $(4\times6)^{-6}$ in the numerator and $(5\times10)^{-6}$ in the denominator.
Step 3: Use the rule $(ab)^n = a^n \times b^n$. Thus, we have: $(4 \times 6)^{-6} = 4^{-6} \times 6^{-6}$ $(5 \times 10)^{-6} = 5^{-6} \times 10^{-6}$ Combining these gives the full expression: $\frac{4^{-6} \times 6^{-6}}{5^{-6} \times 10^{-6}}$

Therefore, the simplified form of the given expression and the correct choice is $\frac{4^{-6} \times 6^{-6}}{5^{-6} \times 10^{-6}}$.

To solve this problem, let's break down the expression and apply the rules of exponents:

Step-by-Step Solution:

• Step 1: Understand that the expression given is $\left(\frac{3 \times 6}{5}\right)^{-4}$.
• Step 2: Simplify the fraction $\frac{3 \times 6}{5}$ as a single fraction, which is already given.
• Step 3: Use the property of negative exponents: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$. This lets us convert the expression.
• Step 4: Apply the negative exponent $-4$ to each component inside the fraction: $(3 \times 6)^{-4}$ becomes $3^{-4} \times 6^{-4}$ and the denominator $5^{-4}$.

Combining these steps results in the expression:

$\frac{3^{-4} \times 6^{-4}}{5^{-4}}$

This matches choice 3, which is the correct answer.

Therefore, the solution to the problem is $\frac{3^{-4} \times 6^{-4}}{5^{-4}}$.

To solve this expression, we need to apply the rules of exponents to simplify $\left(\frac{1}{4 \times 6 \times 9}\right)^{-4}$.

First, using the power of a fraction rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, we express each term separately as follows:

$(4 \times 6 \times 9)^{-4} = 4^{-4} \times 6^{-4} \times 9^{-4}$

Therefore, the original negative exponent transforms to a multiplication of three positive exponents in the denominator represented as:

$\frac{1^{-4}}{4^{-4} \times 6^{-4} \times 9^{-4}}$

Thus, the corresponding expression in terms of powers with negative exponents is:

$\frac{1^{-4}}{4^{-4} \times 6^{-4} \times 9^{-4}}$

The correct answer is Choice 4.

To solve this problem, let's break down the expression $\left(\frac{3}{5 \times 8 \times 7}\right)^{-2}$:

• Step 1: Recognize that we have a fraction raised to a negative exponent.
• Step 2: Apply the power of a fraction rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.
• Step 3: Specifically, apply the rule with a negative exponent: $\left(\frac{3}{5 \times 8 \times 7}\right)^{-2} = \frac{3^{-2}}{(5 \times 8 \times 7)^{-2}}$.
• Step 4: Use the negative exponent rule for each element in the expression $(5 \times 8 \times 7)^{-2} = 5^{-2} \times 8^{-2} \times 7^{-2}$.

This gives us the expression: $\frac{3^{-2}}{5^{-2} \times 8^{-2} \times 7^{-2}}$.

Therefore, the correct expression is $\frac{3^{-2}}{5^{-2} \times 8^{-2} \times 7^{-2}}$.

The given expression is:
$\left(\frac{7\times11\times19}{3\times12\times15}\right)^{-4}$

To solve this expression, we need to apply the rules of exponents, specifically the rule for powers of a fraction. For any fraction$\left(\frac{a}{b}\right)^{-n}$, the expression is equivalent to$\left(\frac{b}{a}\right)^n$.
Therefore, negative exponents indicate that the fraction should be flipped and raised to the positive of that exponent.

Substitute the terms into this formula:
1. Flip the fraction: $\left(\frac{3\times12\times15}{7\times11\times19}\right)$
2. Raise both numerator and denominator to the power of 4:
Thus, we have:
$\left(\frac{3\times12\times15}{7\times11\times19}\right)^{4}$

Now evaluating each term individually:
- In the numerator:
- $3^4\times12^4\times15^4$
- In the denominator:
- $7^4\times11^4\times19^4$

Applying the negative exponent rule, each individual factor in both numerator and denominator should be inverted, altering the exponents to negative:
1. Numerator becomes: $3^{-4}\times12^{-4}\times15^{-4}$
2. Denominator becomes: $7^{-4}\times11^{-4}\times19^{-4}$

Rewriting the expression, we achieve:
$\frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}$

This matches precisely the provided solution.

The solution to the question is:$\frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}$