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Chapter 14: Problem 52

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$\left(x^{\frac{2}{3}}-\frac{1}{\sqrt[3]{x}}\right)^{3}$$

Short Answer

Expert verified
The simplified form of the expression \( \left(x^{2/3}-\frac{1}{\sqrt[3]{x}}\right) ^3 \) is \( x^2 - 3 + 3x^{-2/3} - x^{-2} \)

Step by step solution

01

Identify a, b, and n

Identify the values for a, b and n from the expression. In this case, \( a = x^{2/3} \), \( b = -1/(\sqrt[3]{x}) \) and \( n = 3 \).
02

Apply the Binomial Theorem

Use the Binomial Theorem to expand the expression. The expansion will take the form of: \( (a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^k \). Substituting the values of a, b and n from our expression gives us: \[ (x^{2/3}-1/\sqrt[3]{x})^3 = {3 \choose 0} (x^{2/3})^3 (-1/\sqrt[3]{x})^0 + {3 \choose 1} (x^{2/3})^2 (-1/\sqrt[3]{x})^1 + {3 \choose 2} (x^{2/3})^1 (-1/\sqrt[3]{x})^2 + {3 \choose 3} (x^{2/3})^0 (-1/\sqrt[3]{x})^3 \]
03

Simplify the expression

Simplify each term in the expression by calculating the binomial coefficients and simplifying the powers of x. The result is: \( x^2 - 3 + 3x^{-2/3} - x^{-2} \).

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