Determine whether each of the following expressions is equivalent to x^8/3
8√x3
((3√x)^4)^2
3√x^4 x 3√x^4
(3√x)^0/(3√x^2)^-4
To determine whether each expression is equivalent to x^(8/3), we need to simplify and see if they result in the same value.
1) 8√x^3
This can be rewritten as 8(x^3)^(1/2) = 8√(x^3).
This expression is not equivalent to x^(8/3) since there is an extra x term under the square root.
2) ((3√x)^4)^2
This can be simplified as ((x^(1/3))^4)^2 = (x^(4/3))^2 = x^(8/3).
This expression is equivalent to x^(8/3).
3) 3√x^4 * 3√x^4
This can be simplified as (x^(4/3))*(x^(4/3)) = x^(8/3).
This expression is equivalent to x^(8/3).
4) (3√x)^0 / (3√x^2)^-4
Using the rule that (a^m)^n = a^(m*n), this expression becomes (x^0)/(x^(-8/3)).
Recall that x^0 = 1, and x^(-n) = 1/x^n.
Therefore, this expression simplifies to 1 / (1/x^(8/3)) = x^(8/3).
This expression is equivalent to x^(8/3).
In summary:
- 8√x^3 is not equivalent to x^(8/3).
- ((3√x)^4)^2, 3√x^4 * 3√x^4, and (3√x)^0 / (3√x^2)^-4 are all equivalent to x^(8/3).