Powers of a Fraction - Examples, Exercises and Solutions

Let's determine the corresponding expression for $\left(\frac{2}{3}\right)^a$:

We apply the property of exponentiation for fractions, which states:
$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$.

Substituting $x = 2$, $y = 3$, and $n = a$, we have:

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

Therefore, the correct expression is $\frac{2^a}{3^a}$.

Assessing the possible choices:

• Choice 1: $\frac{2}{3^a}$ - This is incorrect as it does not raise the numerator to $a$.
• Choice 2: $\frac{2a}{3a}$ - This is incorrect as it misuses the exponent rule.
• Choice 3: $\frac{2^a}{3^a}$ - This is correct, as it follows the exponentiation property.
• Choice 4: $\frac{2^a}{3}$ - This is incorrect as it does not raise the denominator to $a$.

Thus, the correct choice is Choice 3: $\frac{2^a}{3^a}$.

The fraction $\frac{10}{13}$ raised to the power of 8 can be expressed by applying the power to both the numerator and the denominator based on the rule for powers of a fraction:

$\left(\frac{10}{13}\right)^8 = \frac{10^8}{13^8}$

To solve for the given expression, we use the formula $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. This means that the fraction power rule allows us to take each component of the fraction and raise it to the required power:

• Step 1: Apply the power of 8 to the numerator: $10^8$.
• Step 2: Apply the power of 8 to the denominator: $13^8$.
• Step 3: Combine both into a single fraction: $\frac{10^8}{13^8}$.

Thus, the expression $\left(\frac{10}{13}\right)^8$ simplifies to $\frac{10^8}{13^8}$.

Therefore, the correct answer from the choices provided is $\frac{10^8}{13^8}$, corresponding to choice 3.

The problem asks us to express $\left(\frac{3}{7}\right)^6$ in another form. To solve this, we apply the exponent rule for fractions: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

• First, identify the numerator and the denominator in the fraction $\frac{3}{7}$.
• We have $a = 3$ and $b = 7$.
• According to the exponent rule, raise both the numerator and the denominator separately to the power of 6:

$\left(\frac{3}{7}\right)^6 = \frac{3^6}{7^6}$

This signifies that each component of the fraction is raised to the power of 6.

To verify, we compare our result with the given choices:

• Option 1: $\frac{3}{7^6}$ does not apply the exponent to the "3".
• Option 2: $\frac{3^6}{7^6}$, matches our derived expression.
• Option 3: $\frac{3^6}{7}$ does not apply the exponent to the "7".
• Option 4: $6\times\left(\frac{3}{7}\right)^5$ changes the power on the entire fraction and multiplies by 6, which is incorrect based on our interpretation.

Therefore, the solution to the problem is $\frac{3^6}{7^6}$, which corresponds to choice 2.

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 $\frac{a^9}{b^9}$.

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 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 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'll apply the rule for raising a fraction to a power:

Using the formula $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, we can express $\left(\frac{5}{8}\right)^9$ as follows:

Step 1: Identify the base and exponent in $\left(\frac{5}{8}\right)^9$. Here, $a = 5$, $b = 8$, and $n = 9$.

Step 2: Apply the exponentiation rule:
$\left(\frac{5}{8}\right)^9 = \frac{5^9}{8^9}$.

Therefore, the original expression simplifies to $\frac{5^9}{8^9}$.

As a result, the correct rewritten form of $\left(\frac{5}{8}\right)^9$ is $\frac{5^9}{8^9}$.

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

• Identify the given expression which is $\left(\frac{20}{21}\right)^4$.

• Apply the exponentiation rule for fractions: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

• Calculate $20^4$ and $21^4$ and place them as the numerator and denominator, respectively.

Now, let's work through each step:
Step 1: We begin with the expression $\left(\frac{20}{21}\right)^4$.
Step 2: Using the power of a fraction rule, we have $\left(\frac{20}{21}\right)^4 = \frac{20^4}{21^4}$.

Therefore, the corresponding simplified expression is $\frac{20^4}{21^4}$.

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

• Step 1: Identify the given expression $\left(\frac{13}{19}\right)^7$.
• Step 2: Apply the exponent rule for fractions: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.
• Step 3: Perform the calculation by raising both the numerator and the denominator to the power of 7.

Now, let's work through each step:
Step 1: The expression provided is $\left(\frac{13}{19}\right)^7$, which is a fraction raised to an exponent.
Step 2: Using the exponentiation rule for fractions: $\left(\frac{a}{b}\right)^n$ is equivalent to $\frac{a^n}{b^n}$.
Step 3: Applying this rule, we express $\left(\frac{13}{19}\right)^7$ as $\frac{13^7}{19^7}$.

Therefore, the solution to the problem is $\frac{13^7}{19^7}$, which corresponds to choice 1.

To solve this problem, we'll use the rule for powers of a fraction, which states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

Given the expression $\left(\frac{2}{9}\right)^7$, we apply this exponent rule:

$\left(\frac{2}{9}\right)^7 = \frac{2^7}{9^7}$

This means we raise the numerator, 2, to the power of 7, and the denominator, 9, also to the power of 7.

The matching choice in the given options is:

• Choice 1: $\frac{2^7}{9^7}$

Therefore, the solution to the problem is $\frac{2^7}{9^7}$.

To solve this problem, we must square the fraction $\left(\frac{1}{2}\right)$. The exponent rule for fractions states that when you raise a fraction $\left(\frac{a}{b}\right)$ to the power of $n$, it becomes $\frac{a^n}{b^n}$.

Here, in the fraction $\left(\frac{1}{2}\right)$, we identify the numerator $a = 1$ and the denominator $b = 2$, with the exponent $n = 2$.

Applying the formula, we calculate the result:
$\left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2}$

Therefore, the expression that corresponds to $\left(\frac{1}{2}\right)^2$ is $\frac{1^2}{2^2}$, which directly matches the given choice.

The correct choice from the answer options is:

• Choice 3: $\frac{1^2}{2^2}$

Therefore, the solution to the problem is $\frac{1^2}{2^2}$.

To solve this problem, we need to simplify the expression $\left(\frac{1}{5}\right)^3$ using the exponent rules for fractions:

• Step 1: Identify the base and the exponent. Here, the base is $\frac{1}{5}$ and the exponent is $3$.
• Step 2: Apply the rule for raising a fraction to a power. The rule states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.
• Step 3: Apply the exponent to both the numerator and the denominator:

Thus, $\left(\frac{1}{5}\right)^3 = \frac{1^3}{5^3}$.

Therefore, the simplified expression is $\frac{1^3}{5^3}$, which corresponds to choice 1 in the provided options.

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

• Step 1: We are given the expression $\left(\frac{2}{3}\right)^2$.
• Step 2: Apply the fraction exponent rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

Applying this rule to our expression:

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

Calculating further would give:

$\frac{2^2}{3^2} = \frac{4}{9}$.

However, the question asks to only match the expression, which is $\frac{2^2}{3^2}$.

The correct choice from the given options is $\frac{2^2}{3^2}$.

This matches Choice 3 in the provided multiple choices.

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:

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