Question

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

Answer 1

To prove the divisibility of 45^45 * 15^15 by 75^30, we rewrite the expression and show that it is divisible.

Step-by-step explanation:

To prove the divisibility of 45^45 * 15^15 by 75^30, we need to show that 75^30 is a factor of 45^45 * 15^15.

We can rewrite 45 as 5 * 9 and 15 as 3 * 5. Thus, 45^45 * 15^15 can be expressed as (5 * 9)^45 * (3 * 5)^15.

Now, we can simplify the expression as (5^45 * 9^45) * (3^15 * 5^15).

Since 75 = 5^2 * 3, we can rewrite 75^30 as (5^2 * 3)^30 = (5^60 * 3^30).

It can be observed that (5^45 * 9^45) * (3^15 * 5^15) is divisible evenly by (5^60 * 3^30), which means 75^30 is a factor of 45^45 * 15^15.

Answer 2

`3^(75)`

Step-by-step explanation:

We are asked to divide our given fraction:
`(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,

`(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,

`(3^(45)*15^(45+15))/(5^(30)*15^(30))`

`(3^(45)*15^(60))/(5^(30)*15^(30))`

Using quotient rule of exponent
`(a^m)/(a^n)=a^(m-n)` we will get,

`(3^(45)*15^(60-30))/(5^(30))`

`(3^(45)*15^(30))/(5^(30))`

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

`(3^(45)*(3*5)^(30))/(5^(30))`

`(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)`.