To solve the expression \((x^{1/3} - x^{-1/3})(x^{2/3} + 1 + x^{-2/3})\), you can use the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\) to simplify it.
Let's break it down step by step:
1. Notice that \((x^{1/3})^3 = x\) and \((x^{-1/3})^3 = x^(-1)\), so you can rewrite \(x^{1/3}\) as \(x^{3/3}\) and \(x^{-1/3}\) as \(x^{-3/3}\).
2. Apply the difference of squares formula to the expression:
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To solve the expression \((x^{1/3} - x^{-1/3})(x^{2/3} + 1 + x^{-2/3})\), you can use the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\) to simplify it.
Let's break it down step by step:
1. Notice that \((x^{1/3})^3 = x\) and \((x^{-1/3})^3 = x^(-1)\), so you can rewrite \(x^{1/3}\) as \(x^{3/3}\) and \(x^{-1/3}\) as \(x^{-3/3}\).
2. Apply the difference of squares formula to the expression:
\((x^{1/3} - x^{-1/3})(x^{2/3} + 1 + x^{-2/3}) = (x^{3/3} - x^{-3/3})(x^{2/3} + 1 + x^{-2/3})\)
3. Now, you have a difference of squares: \(a^2 - b^2\), where \(a = x^{3/3}\) and \(b = x^{-3/3}\). Apply the formula:
\((x^{3/3} - x^{-3/3})(x^{2/3} + 1 + x^{-2/3}) = (x^{3/3} + x^{-3/3})(x^{2/3} + 1 + x^{-2/3})\)
4. Simplify the exponents:
\((x - x^{-1})(x^{2/3} + 1 + x^{-2/3})\)
So, the simplified expression is \((x - x^{-1})(x^{2/3} + 1 + x^{-2/3})\).
Step-by-step explanation:
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