Question

Simplify $\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}.$ $\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}.$

Answer 1

Answer:

$3 -\sqrt[2]3$

Step-by-step explanation:

Given

$\frac{2\sqrt[3]{9}}{1 + \sqrt[3]{3} + \sqrt[3]{9}}$

Required

Simplify

Rewrite the given expression in index form

$\frac{2 * 9 ^\frac{1}{3}}{1 + 3^{\frac{1}{3}} + 9^{\frac{1}{3}}}$

Express 9 as 3²

$\frac{2 * 3^2^*^\frac{1}{3}}{1 + 3^{\frac{1}{3}} + 3^2^*^{\frac{1}{3}}}$

$\frac{2 * 3^\frac{2}{3}}{1 + 3^{\frac{1}{3}} + 3^{\frac{2}{3}}}$

Multiply the numerator and denominator by $1 - 3^{\frac{1}{3}}$

$\frac{2 * 3^\frac{2}{3}}{1 + 3^{\frac{1}{3}} + 3^{\frac{2}{3}}} * \frac{1 - 3^{\frac{1}{3}}}{1 - 3^{\frac{1}{3}}}$

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

Open the bracket

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

Simplify the Numerator using Laws of Indices

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

Further Simplify

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

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

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

Simplify the denominator

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

Further Simplify Using Laws of Indices

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

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

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

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

Collect Like Terms

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

Group Like Terms for Clarity

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

$\frac{2 (3^\frac{2}{3}) -2 (3)}{(- 2)+ (0) + (0)}}$

$\frac{2 (3^\frac{2}{3}) -2 (3)}{-2}}$

Divide the fraction

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

Reorder the above expression

$3 -3^\frac{2}{3}$

The expression can be represented as

$3 -\sqrt[2]3$

Hence;

$\frac{2\sqrt[3]{9}}{1 + \sqrt[3]{3} + \sqrt[3]{9}}$ when simplified is equivalent to $3 -\sqrt[2]3$