Answer:
[tex]3^{75}[/tex].
Step-by-step explanation:
We have been an division problem: [tex]\frac{45^{45}*15^{15}}{75^{30}}[/tex].
We will simplify our division problem using rules of exponents.
Using product rule of exponents [tex](a*b)^n=a^n*b^n[/tex] we can write:
[tex]45^{45}=(9*5)^{45}=9^{45}*5^{45}[/tex]
[tex]15^{15}=(3*5)^{15}=3^{15}*5^{15}[/tex]
[tex]75^{30}=(15*5)^{30}=15^{30}*5^{30}[/tex]
Substituting these values in our division problem we will get,
[tex]\frac{9^{45}*5^{45}*3^{15}*5^{15}}{15^{30}*5^{30}}[/tex]
Using power rule of exponents [tex]a^n*a^m=a^{n+m}[/tex] we will get,
[tex]\frac{9^{45}*5^{(45+15)}*3^{15}}{15^{30}*5^{30}}[/tex]
[tex]\frac{9^{45}*5^{60}*3^{15}}{15^{30}*5^{30}}[/tex]
Using product rule of exponents [tex](a*b)^n=a^n*b^n[/tex] we will get,
[tex]\frac{(3*3)^{45}*5^{60}*3^{15}}{(3*5)^{30}*5^{30}}[/tex]
[tex]\frac{3^{45}*3^{45}*5^{60}*3^{15}}{3^{30}*5^{30}*5^{30}}[/tex]
Using power rule of exponents [tex]a^n*a^m=a^{n+m}[/tex] we will get,
[tex]\frac{3^{(45+45+15)}*5^{60}}{3^{30}*5^{(30+30)}}[/tex]
[tex]\frac{3^{105}*5^{60}}{3^{30}*5^{60}}[/tex]
[tex]\frac{3^{105}}{3^{30}}[/tex]
Using quotient rule of exponent [tex]\frac{a^m}{a^n}=a^{m-n}[/tex] we will get,
[tex]\frac{3^{105}}{3^{30}}=3^{105-30}[/tex]
[tex]3^{105-30}=3^{75}[/tex]
Therefore, our resulting quotient will be [tex]3^{75}[/tex].
