Question

Simplify the following expression as much as you can use exponential properties. (6^-2)(3^-3)(3*6)^4

Answer 1

Simplifying the expression
`(6^(-2))(3^(-3))(3*6)^4` we get
`\mathbf{108}`

Explanation:

We need to simplify the expression
`(6^(-2))(3^(-3))(3*6)^4`

Solving:

`(6^(-2))(3^(-3))(3*6)^4`

Applying exponent rule:
`a^(-m)=(1)/(a^m)`

`=(1)/((6^(2)))(1)/((3^(3)))(18)^4\\=((18)^4)/(6^(2)\:.\:3^(3)) \\`

Factors of
`18=2* 3* 3=2*3^2`

Factors of
`6=2* 3`

Replacing terms with factors

`=((2*3^2)^4)/((2* 3)^(2)\:.\:3^(3)) \\=((2)^4*(3^2)^4)/((2)^2* (3)^(2)\:.\:3^(3)) \\`

Using exponent rule:
`(a^m)^n=a^(m* n)`

`=((2)^4*(3)^8)/((2)^2* (3)^(2)\:.\:3^(3)) \\=(2^4* 3^8)/(2^2* 3^(2)\:.\:3^(3))`

Using exponent rule:
`a^m.a^n=a^(m+n)`

`=(2^4* 3^8)/(2^2* 3^(2+3))\\=(2^4* 3^8)/(2^2* 3^(5))`

Now using exponent rule:
`(a^m)/(a^n)=a^(m-n)`

`=2^(4-2)* 3^(8-5)\\=2^(2)* 3^(3)\\=4* 27\\=108`

So, simplifying the expression
`(6^(-2))(3^(-3))(3*6)^4` we get
`\mathbf{108}`