Answer:
[tex]3^{75}[/tex]
Step-by-step explanation:
We are asked to divide our given fraction: [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}=(3*15)^{45}=3^{45}*15^{45}[/tex]
[tex]75^{30}=(5*15)^{30}=5^{30}*15^{30}[/tex]
Substituting these values in our division problem we will get,
[tex]\frac{3^{45}*15^{45}*15^{15}}{5^{30}*15^{30}}[/tex]
Using power rule of exponents [tex]a^m*a^n=a^{m+n}[/tex] we will get,
[tex]\frac{3^{45}*15^{45+15}}{5^{30}*15^{30}}[/tex]
[tex]\frac{3^{45}*15^{60}}{5^{30}*15^{30}}[/tex]
Using quotient rule of exponent [tex]\frac{a^m}{a^n}=a^{m-n}[/tex] we will get,
[tex]\frac{3^{45}*15^{60-30}}{5^{30}}[/tex]
[tex]\frac{3^{45}*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^{45}*(3*5)^{30}}{5^{30}}[/tex]
[tex]\frac{3^{45}*3^{30}*5^{30}}{5^{30}}[/tex]
Upon canceling out [tex]5^{30}[/tex] we will get,
[tex]3^{45}*3^{30}[/tex]
Using power rule of exponents [tex]a^m*a^n=a^{m+n}[/tex] we will get,
[tex]3^{45+30}[/tex]
[tex]3^{75}[/tex]
Therefore, our resulting quotient will be [tex]3^{75}[/tex].
