Simplify the Complex Fraction: (8⁷ × 13⁶)/(19⁶ × 3⁷)

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