\( \frac{^{3}\sqrt {x^{2}}-^{3}\sqrt {x}}{^{3}\sqrt {x}} \)

To simplify the expression \( \frac{^{3}\sqrt {x^{2}}-^{3}\sqrt {x}}{^{3}\sqrt {x}} \), let's break it down step by step. We know that \( ^{3}\sqrt{x} = x^{1/3} \). Thus, we can rewrite the terms: - \( ^{3}\sqrt{x^{2}} = (x^{2})^{1/3} = x^{2/3} \) - \( ^{3}\sqrt{x} = x^{1/3} \) Substituting these into the original expression, we get: \[ \frac{x^{2/3} - x^{1/3}}{x^{1/3}} \] Now, we can simplify the numerator: \[ x^{2/3} - x^{1/3} = x^{1/3}(x^{1/3} - 1) \] Now, substituting back into the expression gives: \[ \frac{x^{1/3}(x^{1/3} - 1)}{x^{1/3}} \] As long as \( x^{1/3} \neq 0 \) (which implies \( x \neq 0 \)), we can cancel \( x^{1/3} \): \[ x^{1/3} - 1 \] Thus, the simplified form of your original expression is: \[ x^{1/3} - 1 \]