Powers of a Fraction: Combination of different bases

To solve the problem, we will utilize the properties of exponents to express the given fraction properly.

The given expression is $\frac{2^4 \times 3^5}{7^4 \times 11^5}$. Our goal is to rewrite this expression using properties of exponents.

Let's break it down into two separate expressions based on the properties of division of exponents:

• $\frac{2^4}{7^4} = \left(\frac{2}{7}\right)^4$ because $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$.
• $\frac{3^5}{11^5} = \left(\frac{3}{11}\right)^5$ for the same reasons as above.

By rewriting the original expression using these properties, we have:

$\frac{2^4 \times 3^5}{7^4 \times 11^5} = \left(\frac{2}{7}\right)^4 \times \left(\frac{3}{11}\right)^5$.

Therefore, the expression can be rewritten as $\left(\frac{2}{7}\right)^4 \times \left(\frac{3}{11}\right)^5$. This corresponds to choice 3 in the provided options.

The correct transformation of the expression is effectively represented and confirmed with the properties of exponents.

To solve the problem, we apply the properties of exponents and simplify each component of the fraction separately.

Step 1: Analyze the expression.

• The given expression is $\frac{8^7\times13^6}{19^6\times3^7}$.
• We need to simplify this using exponent rules.

Step 2: Apply the rule of exponents to simplify each pair of terms.

• Simplify the expression $\frac{8^7}{3^7}$ using the rule $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ to get $\left(\frac{8}{3}\right)^7$.
• Simplify $\frac{13^6}{19^6}$ to get $\left(\frac{13}{19}\right)^6$.

Step 3: Combine the simplified components into a single expression.

• The expression becomes $\left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6$.

choice 1 : The simplified expression matches choice $\left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6$, which means this is the correct answer.

Therefore, the simplified expression is $\left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6$.