Which of the following conclusions is true about the statement below?

x^2 = √x

a) The statement is always true.

b) The statement is true when x is negative.

c) The statement is true when x=0.

d) The statement is never true.

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To determine which conclusion is true about the statement x^2 = √x, we need to analyze the equation.

First, let's square both sides of the equation: (x^2)^2 = (√x)^2
This simplifies to: x^4 = x

Now, we can consider each conclusion:

a) The statement is always true.
The equation x^4 = x is not always true. For example, when x=0 or x=1, the equation is true, but for other x values, it is not true. Therefore, this conclusion is not true.

b) The statement is true when x is negative.
Substituting a negative value for x in the equation x^4 = x, we get (-x)^4 = -x. If we simplify this, we get x^4 = -x. Therefore, this conclusion is not true.

c) The statement is true when x=0.
When x=0, the equation becomes 0^4 = √0. Simplifying, we get 0 = 0, which is true. Therefore, this conclusion is true.

d) The statement is never true.
Since we found that the statement is true when x=0, this conclusion is not true.

Therefore, the correct conclusion is: c) The statement is true when x=0.

To determine which conclusion is true about the given statement, we need to analyze the equation x^2 = √x.

To make it easier to work with, let's square both sides of the equation:
(x^2)^2 = (√x)^2
x^4 = x

Now, let's rearrange the equation:
x^4 - x = 0

To find the values of x that make this equation true, we need to factor it:
x(x^3 - 1) = 0

Now, we have two possible solutions: either x = 0 or (x^3 - 1) = 0.

If we solve for (x^3 - 1) = 0, we get:
x^3 = 1
x = 1

Therefore, the possible solutions for the equation x^2 = √x are x = 0 and x = 1.

Now, let's go back to the given conclusions and see which one is true.

a) The statement is always true: This is not true because we found that x = 0 and x = 1 are the valid solutions, not all values of x.

b) The statement is true when x is negative: This is not true because neither x = 0 nor x = 1 are negative values.

c) The statement is true when x=0: This is true because if we substitute x = 0 into the equation, we get 0^2 = √0, which is true.

d) The statement is never true: This is not true because we found that x = 0 and x = 1 are valid solutions.

Therefore, the correct conclusion is c) The statement is true when x=0.