The following expression is a rational number. Find its value.
$\sqrt[3]{6 \sqrt{3} + 10} - \sqrt[3]{6 \sqrt{3} - 10}$

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Solution

$6 \sqrt{3} + 10 = {(\sqrt{3} + 1)}^{3} 6 \sqrt{3} - 10 = {(\sqrt{3} - 1)}^{3} \sqrt[3]{6 \sqrt{3} + 10} - \sqrt[3]{6 \sqrt{3} - 10} = (\sqrt{3} + 1) - (\sqrt{3} - 1) = 2$

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Question 6

Which of the following expressions show that rational numbers are associative under multiplication.
(a) $\frac{2}{3} \times (\frac{- 6}{7} \times \frac{3}{5}) = (\frac{2}{3} \times \frac{- 6}{7}) \times \frac{3}{5}$
(b) $\frac{2}{3} \times (\frac{- 6}{7} \times \frac{3}{5}) = \frac{2}{3} \times (\frac{3}{5} \times \frac{- 6}{7})$
(c) $\frac{2}{3} \times (\frac{- 6}{7} \times \frac{3}{5}) = (\frac{3}{5} \times \frac{2}{3}) \times \frac{- 6}{7}$
(d) $(\frac{2}{3} \times \frac{- 6}{7}) \times \frac{3}{5} = (\frac{- 6}{7} \times \frac{2}{3}) \times \frac{3}{5}$

Question 6

Q. The number of rational roots of

{log}_{x}^{3} 10 - 6 {log}_{x}^{2} 10 + 11 {log}_{x} 10 - 6 = 0

, is

Q. Express with rational denominator:

\frac{\sqrt[3]{3}}{\sqrt{3} + \sqrt[6]{9}}

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The following expression is a rational number. Find its value.3√6√3+10−3√6√3−10

6√3+10=(√3+1)36√3−10=(√3−1)33√6√3+10−3√6√3−10=(√3+1)−(√3−1)=2

The following expression is a rational number. Find its value.
$\sqrt[3]{6 \sqrt{3} + 10} - \sqrt[3]{6 \sqrt{3} - 10}$

$6 \sqrt{3} + 10 = {(\sqrt{3} + 1)}^{3} 6 \sqrt{3} - 10 = {(\sqrt{3} - 1)}^{3} \sqrt[3]{6 \sqrt{3} + 10} - \sqrt[3]{6 \sqrt{3} - 10} = (\sqrt{3} + 1) - (\sqrt{3} - 1) = 2$