Question

Please answer the question in the photo.

Answer 1

Answer: Choice B

The equation is sometimes true. When x = 0, 1 or -1, then the equation is true. Otherwise, the equation is false.

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Explanation:

When it comes to something like "always true" all we need is one counter example to break that statement. Pick something like x = 64 and you'll find that the left hand side becomes

$x^{1/3} = 64^{1/3} = \sqrt[3]{64} = \sqrt[3]{4^3} = 4$

while the right hand side turns into

$x^3 = 4^3 = 4*4*4 = 64$

This means the original equation $x^{1/3} = x^3$ updates to $4 = 64$ after plugging in x = 4. We get a false statement making the original statement false when x = 4. So saying  "$x^{1/3} = x^3$ is always true" is not correct.

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The equation is only sometimes true for the x values 0, 1 and -1

Let's try x = 0

$x^{1/3} = x^3\\\\0^{1/3} = 0^3\\\\0 = 0$

we get a true statement confirming x = 0 as a solution

Trying x = 1 leads to

$x^{1/3} = x^3\\\\(1)^{1/3} = 1^3\\\\1 = 1$

also true. A similar situation happens with x = -1 as well, just that we have negatives this time.