Question

Simplify the following exponential expression. Show your work step by step and list the Properties of Exponents used to solve this problem next to your work.

3x^{0}(2x^{3}t^{2})^{4}
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(4x^{7}y^{4})^{2}

Answer 1

Given problem is

`(3x^0(2x^3t^2)^4)/((4x^7y^4)^2)`

distribute outer exponents using formula:
`(a^mb^n)^c=a^(mc)b^(nc)`, we get:

`=(3x^0(2^4x^(3\cdot4)t^(2\cdot4)))/(4^2x^(7\cdot2)y^(4\cdot2))`

Simplify exponents:

`=(3x^0(2^4x^(12)t^8))/(4^2x^(14)y^8)`

plug
`x^0=1`

`=(3\cdot1(2^4x^(12)t^8))/(4^2x^(14)y^8)`

simplify exponents

`=(3(16x^(12)t^8))/(16x^(14)y^8)`

simplify (closure property)

`=(48x^(12)t^8)/(16x^(14)y^8)`

simplify exponent part using formula :
`(a^m)/(a^n)=a^(\left(m-n\right))` we get:

`=(3x^(\left(12-14\right))t^8)/(y^8)`

Simplify exponents:

`=(3x^(\left(-2\right))t^8)/(y^8)`

send term to denominator to avoid negative exponent

`=(3t^8)/(x^2y^8)`

Hence final answer is
`(3t^8)/(x^2y^8)`.

Answer 2

`(3x^0(2x^3t^2)^4)/((4x^7y^4)^2) = (3(1)(2)^4(x^3)^4(t^2)^4)/((4)^2(x^7)^2(y^4)^2)` Since,
`a^0 = 1` and
`(ab)^m=a^mb^m`

`(3x^0(2x^3t^2)^4)/((4x^7y^4)^2) = (3(16)x^(12)t^(8))/(16x^(14)y^8)` Since,
`(a^b)^c=a^(bc)`

`(3x^0(2x^3t^2)^4)/((4x^7y^4)^2) = (3t^(8))/(x^(14-12)y^8)`

`(3x^0(2x^3t^2)^4)/((4x^7y^4)^2) = (3t^(8))/(x^(2)y^8)`

Thus,

`(3x^0(2x^3t^2)^4)/((4x^7y^4)^2) = (3t^(8))/(x^(2)y^8)`