Classify each statement as sometimes, always, or never true.
A. 2^(2/3)*4^(4/3)=8*2^(1/3)
B. If x>0, then (squareroot x*squareroot 2)^3 =x^(3/2) 2^(3/2)
C. If x>0, then (squareroot x+ squareroot 2)^3 =x^(3/2) 2^(3/2)
D. The only solution to x^(2/3)= 4 is x=8
To classify each statement as sometimes, always, or never true, we will analyze each statement individually.
A. 2^(2/3)*4^(4/3) = 8*2^(1/3)
To determine if this statement is sometimes, always, or never true, we can simplify both sides of the equation.
Using exponent rules, we can rewrite 4^(4/3) as (2^2)^(4/3) = 2^(8/3).
Now the equation becomes: 2^(2/3) * 2^(8/3) = 8 * 2^(1/3).
By using the rule of multiplying exponents with the same base, we can combine the terms on both sides:
2^(2/3 + 8/3) = 8 * 2^(1/3).
2^(10/3) = 8 * 2^(1/3).
Since 2^(10/3) = (2^(1/3))^10, we have:
(2^(1/3))^10 = 8 * 2^(1/3).
Now we can rewrite 8 as 2^3:
(2^(1/3))^10 = 2^3 * 2^(1/3).
Using the rule of multiplying exponents with the same base, we get:
2^(10/3) = 2^(3 + 1/3).
By comparing the exponents, we can conclude that the left and right sides are equal.
Therefore, the statement A is always true.
B. If x > 0, then (squareroot x * squareroot 2)^3 = x^(3/2) 2^(3/2)
We can simplify both sides of the equation to determine its truthfulness.
On the left side, we have (square root x * square root 2)^3, which simplifies to (sqrt(x))^3 * (sqrt(2))^3.
This further simplifies to x^(3/2) * 2^(3/2).
On the right side, we have x^(3/2) * 2^(3/2).
Both sides of the equation are the same, resulting in a true statement.
Therefore, statement B is always true.
C. If x > 0, then (squareroot x + squareroot 2)^3 = x^(3/2) 2^(3/2)
Again, we can simplify both sides of the equation to determine its truthfulness.
On the left side, we have (sqrt(x) + sqrt(2))^3, which can be expanded using the binomial expansion or the formula for (a + b)^3.
Using the binomial expansion formula, we get (sqrt(x))^3 + 3 * (sqrt(x))^2 * sqrt(2) + 3 * sqrt(x) * (sqrt(2))^2 + (sqrt(2))^3.
This simplifies to x * sqrt(x) + 3 * x * sqrt(2) + 3 * sqrt(2) * 2 + 2 * sqrt(2).
Simplifying further, we get x * sqrt(x) + 3 * x * sqrt(2) + 6 * sqrt(2) + 2 * sqrt(2).
On the right side, we have x^(3/2) * 2^(3/2).
These two sides of the equation are not equal.
Therefore, statement C is never true.
D. The only solution to x^(2/3) = 4 is x=8.
To determine the truthfulness of this statement, we can solve the equation.
Taking the cube root of both sides, we get (x^(2/3))^(3/2) = 4^(3/2).
Simplifying, we have x^1 = 8.
Therefore, x = 8 is indeed a solution.
Since there is only one solution, the statement D is true.
In summary:
A. The statement is always true.
B. The statement is always true.
C. The statement is never true.
D. The statement is true.