Page:Elementary algebra (1896).djvu/195

Rule. (1) Find the root of the coefficient by Arithmetic, and prefix the proper sign found by Art. 42.

(2) Divide the exponent of every factor of the expression by the index of the proposed root.

Examples

(1)	${\sqrt {3}}\left(-64x^{6}\right)=-4x^{2}$

(2)	${\sqrt {4}}(16a^{8})=2a^{2}.$

(3)	$\left({\frac {81x^{1}0}{25c^{4}}}\right)={\frac {9x^{5}}{5c^{2}}}$

It will be seen that in the last case we operate separately upon, the numerator and the denominator.

EXAMPLES XXI a.

Write the square root of each of following expressions :

1. $4a^{2}b^{4}.$

2. $9x^{6}y^{2}.$

3. $25x^{4}y^{6}.$

4. $16a^{4}b^{2}c^{6}.$

5. $81a^{5}b^{8}.$

6. $100x^{8}.$

7. $a^{2}0b^{1}6c64$

8. $a^{8}b^{2}c^{1}2.$

9. ${\frac {324x^{12}}{169y^{6}}}.$

10. ${\frac {81a^{18}}{36b^{12}}}.$

11. ${\frac {256x^{2}y^{4}}{289p^{14}}}$

12. ${\frac {400a^{40}b^{20}}{81x^{10}y^{18}}}$

Write the cube root of each of the following expressions:

13. $27a^{6}b^{3}c^{3}.$

14. $-8a^{12}b^{9}.$

15. $64x^{6}y^{3}z^{12}.$

16. $-343a^{12}b^{18}$

17. ${\frac {x^{12}y^{9}}{125}}$

18. ${\frac {8x^{9}}{729y^{1}5}}$

19. ${\frac {125a^{3}b^{6}}{216x^{6}y^{9}}}$.

20. $-{\frac {27x^{27}}{64y^{63}}}$.

21. $-{\frac {343x^{15}}{512z^{21}}}$.

Write the value of each of the following expressions:

22. ${\sqrt {6}}\left(729a^{18}b^{6}\right).$

23. ${\sqrt {8}}\left(256a^{8}x^{64}\right).$

24. ${\sqrt {5}}\left(-x^{10}y^{15}\right).$

25. ${\sqrt {7}}{\frac {128}{a^{63}b^{56}}}$

26. ${\sqrt {10}}{\frac {a^{30}x^{50}}{b^{100}}}.$

27. ${\sqrt {9}}{\frac {a^{18}}{b^{27}c^{36}}}$

28. ${\sqrt {11}}{\frac {x^{11}y^{44}}{z^{121}}}$

EVOLUTION OF MULTINOMINALS.

198. The Square Root of Any Multinomial. Since the square of $a+b$ is $a^{2}+2ab+b^{2}$, we have to discover a process by which $a$ and $b$, the terms of the root, can be found when a- -{- 2 ab -- Ir is given. The first term, $a$, is the square root of $a^{2}$.

Arrange the terms according to powers of one letter $a$. The first term is $a^{2}$ and its square root is $a$. Set this