Solve the terms in parentheses first. We'll start on the denominator.
The denominator has an exponent for a fraction that also includes exponents. To multiply exponents within parentheses that are raised to a power, use this rule:
[tex](x^a)^b = x^{a \cdot b}[/tex]
Simplify the denominator:
[tex](\frac{3^4}{7^3} )^2 = \frac{3^8}{7^6}[/tex]
Solve the fractions in the numerator:
[tex](\frac{3}{5})^5 = \frac{3^5}{5^5}[/tex]
[tex](\frac{9}{7})^2} = \frac{9^2}{7^2}[/tex]
The problem should now read:
[tex]\frac{\frac{3^5}{5^5} \cdot \frac{9^2}{7^2}}{\frac{3^8}{7^6}}[/tex]
There is a denominator in a denominator. We can bring that to the numerator of the overall fraction:
[tex]\frac{\frac{3^5}{5^5} \cdot \frac{9^2}{7^2}}{\frac{3^8}{7^6}} = \frac{\frac{3^5}{5^5} \cdot \frac{9^2}{7^2} \cdot {7^6}}{3^8}}[/tex]
Using a calculator, simplify the numerator:
[tex]\frac{3^5}{5^5} \cdot \frac{9^2}{7^2} \cdot {7^6} = \frac{19683}{7}[/tex]
The fraction should now read:
[tex]\frac{\frac{19683}{7}}{3^8}[/tex]
There is a denominator in the numerator. This can be brought down to the overall denominator:
[tex]\frac{\frac{19683}{7}}{3^8} = \frac{19683}{3^8 \cdot 7}[/tex]
Factor 19683:
[tex]19683 = 3 \cdot3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 3^9[/tex]
[tex]\frac{19683}{3^8 \cdot 7} = \frac{3^9}{3^8 \cdot 7}[/tex]
Simplify the exponents:
[tex]\frac{3^9}{3^8} = 3[/tex]
The following fraction will be your answer:
[tex]\boxed{\frac{3}{7}}[/tex]
