Elements of Algebra

CHAP. XIX. OF ALGEBRA. 59 5. Reduce $\sqrt{48}$ to its simplest form. Ans. $4\sqrt{3}$. 6. Reduce $\sqrt{(a^3x - a^2x^2)}$ to its simplest form. Ans. $a\sqrt{(ax - x^2)}$. 7. Reduce $\sqrt[3]{\frac{27a^3b^3}{8b-8a}}$ to its simplest form. Ans. $\frac{3ab}{2}\sqrt[3]{\frac{1}{b-a}}$. 8. Add $\sqrt{6}$ to $2\sqrt{6}$; and $\sqrt{8}$ to $\sqrt{50}$. Ans. $3\sqrt{6}$; and $7\sqrt{2}$. 9. Add $\sqrt{4a}$ and $\sqrt[4]{a^6}$ together. Ans. $(a+2)\sqrt{a}$. 10. Add $\sqrt[3]{\frac{b^2}{c}}$ and $\sqrt[3]{\frac{c^2}{b}}$ together. Ans. $\frac{b^2+c^2}{b\sqrt[3]{bc}}$. 11. Subtract $\sqrt{4a}$ from $\sqrt[4]{a^6}$. Ans. $(a-2)\sqrt{a}$. 12. Subtract $\sqrt[3]{\frac{c^2}{b}}$ from $\sqrt[3]{\frac{b^2}{c}}$. Ans. $\frac{b^2-c^2}{b}\sqrt[3]{\frac{1}{bc}}$. 13. Multiply $\sqrt{\frac{2ab}{3c}}$ by $\sqrt{\frac{9ad}{2b}}$. Ans. $\frac{3a\sqrt{d}}{c}$. 14. Multiply $\sqrt[3]{d}$ by $\sqrt[3]{ab}$. Ans. $\sqrt[3]{(a^2b^2d^3)}$. 15. Multiply $\sqrt{(4a-3x)}$ by $2a$. Ans. $\sqrt{(16a^3 - 12a^2x)}$. 16. Multiply $\frac{a}{2b}\sqrt{(a-x)}$ by $(c-d)\sqrt{ax}$. Ans. $\frac{ac-ad}{2b}\sqrt{(a^2x - ax^2)}$. 17. Divide $a^{\frac{1}{2}}$ by $a^{\frac{1}{4}}$; and $a^{\frac{m}{n}}$ by $a^{\frac{c}{d}}$. Ans. $a^{\frac{1}{4}}$; and $a^{\frac{md-cn}{nd}}$. 18. Divide $\frac{ac-ad}{2b}\sqrt{(a^2x-ax^2)}$ by $\frac{a}{2b}\sqrt{(a-x)}$. Ans. $(c-d)\sqrt{ax}$. 19. Divide $a^2 - ad - b + d\sqrt{b}$ by $a-\sqrt{b}$. Ans. $a+\sqrt{b}-d$. 20. What is the cube of $\sqrt[3]{2}$? Ans. $\sqrt[3]{8}$. 21. What is the square of $3\sqrt[3]{bc}$? Ans. $9\sqrt[3]{b^2c^2}$. 22. What is the fourth power of $\frac{a}{2b}\sqrt{\frac{2a}{c-b}}$? Ans. $\frac{a^6}{4b^4(c^2-2bc+b^2)}$. 23. What is the square of $3+\sqrt{5}$? Ans. $14+6\sqrt{5}$. 24. What is the square root of $a^3$? Ans. $a^{\frac{3}{2}}$; or $\sqrt{a^3}$. 25. What is the cube root of $\sqrt{(a^2-x^2)}$? Ans. $\sqrt[6]{(a^2-x^2)}$.