Math  /  Calculus

Question Find the product and ratio of f(x)=x1/2f(x)=x^{1/2} and g(x)=2x3+1g(x)=\sqrt[3]{2x}+1, and rationalize the denominator of the ratio.

Studdy Solution
Now we have the simplified form of f(x)g(x)\frac{f(x)}{g(x)} with a rationalized denominator.
f(x)g(x)=x12(2x31)223x231\frac{f(x)}{g(x)} = \frac{x^{\frac{1}{2}}(\sqrt[3]{2x}-1)}{2^{\frac{2}{3}}x^{\frac{2}{3}}-1}
a) The product f(x)g(x)f(x) \bullet g(x) is 213x56+x122^{\frac{1}{3}}x^{\frac{5}{6}} + x^{\frac{1}{2}}. b) The quotient f(x)g(x)\frac{f(x)}{g(x)} with a rationalized denominator is x12(2x31)223x231\frac{x^{\frac{1}{2}}(\sqrt[3]{2x}-1)}{2^{\frac{2}{3}}x^{\frac{2}{3}}-1}.

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