Question 1093853
i see this problem as:


(4x^(-1))^2 * (2x^3)^3


this is equal to 4^2 * x^(-2) * 2^3 * x^9


rearrange the terms to get 4^2 * 2^3 * x^(-2) * x^9


simplify further to get 128 * x^(-2) * x^9


combine like terms to get 128 * x^7


that's your solution.


you can confirm using your calculator by evaluating the original expression and the final expression using a random value of x.


if the answer comes out the same, then you did good.


i used x = 25 and i got both expressions equal to 7.8125 * 10^11, so i think i did good.


breaking it down, this is what was involved.


(4x^-1)^2 is equal to 4^2 * (x^-1)^2 which is equal to 4^2 * x^(-1*2) which is equal to 4^2 * x^(-2)


the exponent arithmetic properties that are involved are:


(x^a)^b is equal to x^(a*b).


(x^a * y^b) ^ c is equal to x^(a*c) * y^(b*c).


the same properties apply to (2x^3)^3.


that is equal to 2^3 * (x^3)^3 which is equal to 2^3 * x^(3*3) which is equal to 2^3 * x^9.


you were then left with:


4^2 * x^(-2) * 2^3 * x^9


the other property of exponents involved is:


x^a * x^b is equal to x^(a + b) and:


x^a / x^b is equal to x^(a-b)


there's another property that states:


x^(-a) is equal to 1/x^a.


your expression of 4^2 * x^(-2) * 2^3 * x^9 becomes:


4^2 * 2^3 * x^9 * x^-2)


simplifying and using the properties of exponents, you get:


4^2 * x^(-2) * 2^3 * x^9 becomes 16 * x^(9-2) * 8 which becomes 128 * x^7.


here's some good references on exponent arithmetic that you might find useful.


<a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut26_exp.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut26_exp.htm</a>


<a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut23_exppart1.htm" target= "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut23_exppart1.htm</a>


<a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut24_exppart2.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut24_exppart2.htm</a>


let me know if you have any further questions regarding this.