Powers of a Fraction: Variable in the base of the power

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 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 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:

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

To solve this problem, we will rewrite the expression $\left(\frac{6}{x}\right)^3$ using the power of a fraction rule. The steps are as follows:

• Identify the fraction's numerator $6$ and denominator $x$.

• According to the power of a fraction rule, apply the power 3 to both the numerator and the denominator:

• $\left(\frac{6}{x}\right)^3 = \frac{6^3}{x^3}$.

Therefore, the expression is correctly written as $\frac{6^3}{x^3}$.

Comparing with the provided answer choices, the correct choice is choice $2$:

$\frac{6^3}{x^3}$

We need to rewrite the expression $\left(\frac{a}{3}\right)^2$ using the rule of exponents for fractions. This rule states that if you have a fraction $\left(\frac{m}{n}\right)$ and you raise it to a power $k$, it is equivalent to raising both the numerator and the denominator to the power $k$. Therefore, we have:

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

Here, $a^2$ is the numerator and $3^2$ is the denominator. The expression simplifies to:

$\frac{a^2}{9}$

Based on the provided choices, the correct answer is:

Choice 1: $\frac{a^2}{3^2}$

Therefore, the solution to the given problem is $\frac{a^2}{3^2}$.

To solve this problem, our goal is to express the given quotient of powers in a simplified form using exponent laws.

• Step 1: Understand the original expression. We have $\frac{a^5 \times x^5}{7^5 \times b^5}$.
• Step 2: Recognize the structure. Notice that both the numerator and denominator are raised to the fifth power.
• Step 3: Apply the property of exponents for quotients and products, which states that $\left(\frac{m}{n}\right)^k = \frac{m^k}{n^k}$ and $(m \cdot n)^k = m^k \cdot n^k$.
• Step 4: Rewrite the expression as a single fraction raised to the power of 5. Since each term in the numerator and denominator is raised to the fifth power separately, we combine them under a single power:
• Numerator: $a^5 \times x^5 = (a \times x)^5$
• Denominator: $7^5 \times b^5 = (7 \times b)^5$
• Therefore, $\frac{a^5 \times x^5}{7^5 \times b^5} = \left(\frac{a \times x}{7 \times b}\right)^5$.

Thus, the expression can be written as: $\left(\frac{a \times x}{7 \times b}\right)^5$.

Now, comparing this with the answer choices provided:

• Choice 1: $\frac{(a \times x)^5}{7 \times b^5}$ - does not match, as it retains the separate powers incorrectly.
• Choice 2: $\left(\frac{a \times x}{7 \times b}\right)^5$ - matches perfectly as derived.
• Choice 3: $\frac{a \times x^5}{(7 \times b)^5}$ - incorrect form compared to derived structure.
• Choice 4: $7 \times \left(\frac{a \times x}{b}\right)^5$ - unrelated format, doesn't match.

The correct choice is therefore Choice 2. This matches our derived expression using the laws of exponents correctly.

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: $\frac{1}{a^{-x}} = a^x$.