Powers of Fractions Practice Problems and Worksheets

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

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.

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'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.