MathematicsA rationalizing factor of 253+[1253]−1 is

A rationalizing factor of $\sqrt[3]{25} + [\frac{1}{\sqrt[3]{25}}] - 1$ is

A
$5^{\frac{1}{3}} - 5^{\frac{- 1}{3}}$
B
$5^{\frac{1}{3}} + 5^{\frac{- 1}{3}}$
C
$25^{\frac{1}{3}} + 25^{\frac{- 1}{3}}$
D
$25^{\frac{1}{3}} - 25^{\frac{- 1}{3}}$

Solution:

25^{\frac{1}{3}} + \frac{1}{25^{\frac{1}{3}}} - 1

${(5^{\frac{1}{3}})}^{2} - 5^{\frac{1}{3}} \times 5^{\frac{- 1}{3}} + {(5^{\frac{- 1}{3}})}^{2}$

$(a + b) (a^{2} - a b + b^{2}) = a^{3} + b^{3}$

$∴$ rationalizing factor of $25^{\frac{1}{3}} + \frac{1}{25^{\frac{1}{3}}} - 1$ is $5^{\frac{1}{3}} + 5^{\frac{- 1}{3}}$

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