Determine whether each of the following expressions is equivalent to x ^ (8/3)
((root(x, 3)) ^ 4) ^ 2 Yes or No
((root(x, 3)) ^ 0)/((root(x ^ 2, 3)) ^ - 4) Yes or No
root(x ^ 4, 3) * root(x ^ 4, 3) Yes or No
root(x ^ 3, 8) Yes or No
To determine whether each of the given expressions is equivalent to x^(8/3), we can simplify them and see if they match the simplified form.
1. ((root(x, 3)) ^ 4) ^ 2:
Using the rule (a^b)^c = a^(b * c), we simplify the expression as (root(x, 3))^(4 * 2) = (root(x, 3))^8.
Since 8/3 is not equivalent to 8, this expression is NOT equivalent to x^(8/3). Therefore, the answer is No.
2. ((root(x, 3)) ^ 0)/((root(x^2, 3)) ^ -4):
Since any non-zero number raised to the power of 0 is 1, the numerator simplifies to 1.
Using the rule (a^b)^c = a^(b * c), we simplify the denominator as (root(x^2, 3))^(-4 * -1) = (root(x^2, 3))^4.
Therefore, the expression can be simplified to 1/(root(x^2, 3))^4.
Since 8/3 is not equivalent to 4, this expression is NOT equivalent to x^(8/3). Therefore, the answer is No.
3. √(x^4) * √(x^4):
Using the rule √(a * b) = √a * √b, we can rewrite the expression as √(x^4 * x^4) = √(x^8).
Since 8/3 is equivalent to 8/3, this expression is equivalent to x^(8/3). Therefore, the answer is Yes.
4. ∛(x^3, 8):
The expression ∛(x^3, 8) represents the cube root of x^3 in the base of 8.
Using the rule ∛(a^b, c) = a^(b/c), we can rewrite the expression as x^(3/8).
Since 8/3 is not equivalent to 3/8, this expression is NOT equivalent to x^(8/3). Therefore, the answer is No.