Powers of Fractions Practice Problems and Worksheets

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