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

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: $\frac{2^a}{3^a}$.

To solve this problem, we will apply the rules of exponents:

• Step 1: Identify the Given Expression
• Step 2: Apply the Exponent Rule

Now, let's work through these steps:

Step 1: We are given the expression $\left(\frac{3}{4}\right)^x$.

Step 2: According to the rule of exponents, when a fraction is raised to a power, this is equivalent to raising both the numerator and the denominator to that power. Therefore, we have:

$\left(\frac{3}{4}\right)^x = \frac{3^x}{4^x}$

Therefore, the expression $\left(\frac{3}{4}\right)^x$ is equivalent to $\frac{3^x}{4^x}$.

Thus, the correct answer is option 1, which is $\frac{3^x}{4^x}$.

The solution to the problem is $\frac{3^x}{4^x}$.

To solve the problem, follow these steps:

• Identify the given expression: $\left(\frac{5}{7}\right)^{ax}$.
• Apply the rule for exponentiation of a fraction: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.
• Exponentiate both the numerator and the denominator separately by $ax$:
• Calculate $\frac{5^{ax}}{7^{ax}}$ which follows directly from the application of the property.

Therefore, the rewritten expression is $\frac{5^{ax}}{7^{ax}}$.

Among the given choices, the correct one is:

• Choice 4: $\frac{5^{ax}}{7^{ax}}$

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

Step 1: Recognize the given expression: $\left(\frac{5}{9}\right)^{2x+1}$.

Step 2: Apply the exponent rule to rewrite the expression. According to the rule, this becomes:

$\left(\frac{5}{9}\right)^{2x+1} = \frac{5^{2x+1}}{9^{2x+1}}$.

Therefore, the expression can be rewritten as $\frac{5^{2x+1}}{9^{2x+1}}$.

Among the given choices, the correct option is choice 1: $\frac{5^{2x+1}}{9^{2x+1}}$.

Thus, the solution to the problem is $\frac{5^{2x+1}}{9^{2x+1}}$.

To solve this problem, we need to rewrite the expression $\left(\frac{11}{19}\right)^{a+3b}$ using the rules for powers of fractions. Specifically, we apply the exponent to both the numerator and the denominator separately.

According to the rule $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$, we apply the exponent $a+3b$ to both 11 and 19:

$\left(\frac{11}{19}\right)^{a+3b} = \frac{11^{a+3b}}{19^{a+3b}}$

Therefore, the expression can be rewritten as $\frac{11^{a+3b}}{19^{a+3b}}$, matching choice 3 in the provided options.

Hence, the solution to the problem is $\frac{11^{a+3b}}{19^{a+3b}}$.

Let's analyze the expression we are given:

$\left(\frac{2\times4\times6}{7\times8\times9}\right)^{3x}$

The expression is a power of a fraction. The rule for powers of a fraction is that each component of the fraction must be raised to the power separately. This can be expressed as:

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

Applying this rule to our expression, we have:

• The numerator inside the power: $2 \times 4 \times 6$
• The denominator inside the power: $7 \times 8 \times 9$

Therefore, raising each part to the power $3x$ gives us:

$\frac{(2\times4\times6)^{3x}}{(7\times8\times9)^{3x}}$

Thus, the simplified expression for the given equation is:

$\frac{\left(2\times4\times6\right)^{3x}}{\left(7\times8\times9\right)^{3x}}$

The solution to the question is: $\frac{\left(2\times4\times6\right)^{3x}}{\left(7\times8\times9\right)^{3x}}$

Let's start by examining the expression given in the question:

$\left(\frac{13}{7\times6\times3}\right)^{x+y}$

This expression is a power of a fraction. There is a general rule in exponents which states:

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

Using this rule, we will apply it to our original expression.

Given, $a = 13$, $b = 7\times6\times3$, and $n = x+y$, we can rewrite our expression as:

$\frac{13^{x+y}}{(7\times6\times3)^{x+y}}$

The solution to the question is:

$\frac{13^{x+y}}{(7\times6\times3)^{x+y}}$

To solve the problem, we need to simplify the expression $\left(\frac{11\times9}{10\times12}\right)^{x+a}$ and write it in the form requested in the question.

We begin by using the exponent rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$. Applying this rule here:

$<span class="katex"> \left(\frac{11\times9}{10\times12}\right)^{x+a} = \frac{(11\times9)^{x+a}}{(10\times12)^{x+a}} </span>$

Next, we can simplify the expression further by applying the power over a product rule: $(ab)^n = a^n \times b^n$.

Applying this rule to both the numerator and denominator gives us:

Numerator: $(11\times9)^{x+a} = 11^{x+a} \times 9^{x+a}$

Denominator: $(10\times12)^{x+a} = 10^{x+a} \times 12^{x+a}$

Therefore, the entire expression becomes:

$<span class="katex"> \frac{11^{x+a} \times 9^{x+a}}{10^{x+a} \times 12^{x+a}} </span>$

This matches the given answer. Thus, the solution to the question is:

$\frac{11^{x+a}\times9^{x+a}}{10^{x+a}\times12^{x+a}}$

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