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LenaWriter [7]
3 years ago
5

Prove divisibility 45^45·15^15 by 75^30

Mathematics
1 answer:
Anuta_ua [19.1K]3 years ago
8 0

Answer:

3^{75}

Step-by-step explanation:

We are asked to divide our given fraction: \frac{45^{45}*15^{15}}{75^{30}}.

We will simplify our division problem using rules of exponents.

Using product rule of exponents (a*b)^n=a^n*b^n we can write:

45^{45}=(3*15)^{45}=3^{45}*15^{45}

75^{30}=(5*15)^{30}=5^{30}*15^{30}

Substituting these values in our division problem we will get,

\frac{3^{45}*15^{45}*15^{15}}{5^{30}*15^{30}}

Using power rule of exponents a^m*a^n=a^{m+n} we will get,

\frac{3^{45}*15^{45+15}}{5^{30}*15^{30}}

\frac{3^{45}*15^{60}}{5^{30}*15^{30}}

Using quotient rule of exponent \frac{a^m}{a^n}=a^{m-n} we will get,

\frac{3^{45}*15^{60-30}}{5^{30}}

\frac{3^{45}*15^{30}}{5^{30}}

Using product rule of exponents (a*b)^n=a^n*b^n we will get,

\frac{3^{45}*(3*5)^{30}}{5^{30}}

\frac{3^{45}*3^{30}*5^{30}}{5^{30}}

Upon canceling out 5^{30} we will get,

3^{45}*3^{30}

Using power rule of exponents a^m*a^n=a^{m+n} we will get,

3^{45+30}

3^{75}

Therefore, our resulting quotient will be 3^{75}.

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