Question

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

• A. $6$
• B. $5$
• C. $\frac{26}{5}$
• D. $\frac{41}{5}$

Answer 1

Answer: (C)

$\frac{26}{5}$

$(5^{\frac{1}{3}}+5^{-\frac{1}{3}})(5^{\frac{2}{3}}-1+5^{-\frac{2}{3}})=\left(5^{\frac{1}{3}}\right)(5^{\frac{2}{3}}-1+5^{-\frac{2}{3}})+5^{-\frac{1}{3}}(5^{\frac{2}{3}}-1+5^{-\frac{2}{3}})$
$=(5^{\frac{2}{3}+\frac{1}{3}}-5^{\frac{1}{3}}+5^{-\frac{2}{3}+\frac{1}{3}})+(5^{\frac{2}{3}-\frac{1}{3}}-5^{-\frac{1}{3}}+5^{-\frac{2}{3}-\frac{1}{3}})$
$=5^1-5^{\frac{1}{3}}+5^{-\frac{1}{3}}+5^{\frac{1}{3}}-5^{-\frac{1}{3}}+5^{-1}$
$=5^1+5^{-1}$

$=5+\frac{1}{5}=\frac{25+1}{5}=\frac{26}{5}$