Question

100 POINTS!!!!! PLEASE HELP!

Answer 1

`a^(12)\:b^(4)`

Explanation:

Given expression:

`\left(a^(-4)\:b^(-1)\:c\right)^(-2)\left(a^2\:b\:c\right)^2`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`\implies a^((-4 * -2))\:b^((-1 * -2))\:c^(-2)\:a^((2 * 2))\:b^2\:c^2`

Simplify:

`\implies a^(8)\:b^(2)\:c^(-2)\:a^(4)\:b^2\:c^2`

Collect like terms:

`\implies a^(8)a^(4)\:b^(2)b^2\:c^(-2)c^2`

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies a^((8+4))\:b^((2+2))\:c^((-2+2))`

Simplify:

`\implies a^(12)\:b^(4)\:c^(0)`

`\textsf{Apply exponent rule} \quad a^0=1:`

`\implies a^(12)\:b^(4)(1)`

`\implies a^(12)\:b^(4)`

Answer 2

`a^(12) b^(4)`

Explanation:

To simplify we will have to use the negative exponent rule and the power rule along with some algebra.

Negative Exponent Rule

`a^(-b) =(1)/(a^b)`

Power Rule

`(a^b)^(c) =a^(bc)`

Given

`(a^(-4)b^(-1)c )^(2) (a^2bc)^(2)`

Rewrite
`a^(-4)` using negative exponent rule.

`((1)/(a^(4))* b^(-1)c )^(2) (a^2bc)^(2)`

Rewrite
`b^(-1)` using negative exponent rule.

`((1)/(a^(4))* (1)/(b)*c )^(2) (a^2bc)^(2)`

Simplify

`((c)/(a^4b) )^(2) (a^2bc)^(2)`

Rewrite the base as its reciprocal.

`((a^4b)/(c) )^(2) (a^2bc)^(2)`

Apply the power rule.

`(a^8b^2)/(c^2) *(a^2bc)^(2)`

Apply the power rule.

`(a^8b^2)/(c^2) *a^4b^2c^2`

Cancel the common factor of
`c^2`.

`a^8b^2 a^4b^2`

Apply the power rule.

`a^(12) b^(4)`