Page:Elementary algebra (1896).djvu/188

Hence we obtain the following rule for raising a simple expression to any proposed power:

Rule. (1) Raise the coefficient to the required power by Arithmetic, and prefix the proper sign found by Art. 42.

(2) Multiply the index of every factor of the expression by the exponent of the power required.

Examples.	(1)	$\left(-2x^{2}\right)^{5}$	$=$	$-32x^{10}$.
	(2)	$\left(-3ab^{3}\right)^{6}$	$=$	$729a^{6}b^{18}$.
	(3)	$\left({\frac {2ab^{3}}{3x^{y}}}\right)^{4}$	$=$	${\frac {16a^{4}b^{12}}{81x^{8}y^{4}}}$.

It will be seen that in the last case the numerator and the denominator are operated upon separately.

EXAMPLES XX. a.

Write the square of each of the following expressions:

1. $3ab^{3}$

2. $5x^{2}y^{5}$

3. $-2abc^{2}$

4. $11b^{2}c^{3}$

5. $4xyz^{3}$

6. $-{\frac {2}{3}}a^{2}b^{3}$

7. ${\frac {2x^{2}}{3y^{3}}}$

8. $-{\frac {4}{3x^{2}y}}$

9. $-{\frac {7ab}{3}}$

10. ${\frac {3a^{2}b^{3}}{4c^{5}x^{4}}}$

11. $-2xy^{2}$

12. ${\frac {3a^{5}}{x^{3}}}$

Write the cube of each of the following expressions:

13. $2ab^{2}$

14. $3x^{3}$

15. $-2a^{7}c^{2}$

16. $-3a^{3}b$

17. ${\frac {1}{3y^{2}}}$

18. $-{\frac {3a^{5}}{5a^{3}}}$

19. $7x^{3}y^{4}$

20. ${\frac {2}{3}}a^{5}$

Write the value of each of the following expressions:

21. $\left(3a^{2}b^{3}\right)^{4}$

22. $\left(-a^{2}x\right)^{6}$

23. $\left(-2x^{3}y\right)^{5}$

24. $\left({\frac {3x^{4}}{2y^{3}}}\right)^{7}$

25. $\left({3x^{4}}{2y^{3}}\right)^{5}$

26. $\left({\frac {2x^{3}}{3y}}\right)^{8}$

27. $\left(-{\frac {x^{3}}{3}}\right)^{7}$

28. $\left(-{\frac {2x^{5}}{3a^{4}}}\right)^{6}$

29. $\left(-{\frac {2a^{2}x^{3}}{5bc^{2}}}\right)^{4}$

TO SQUARE A BINOMIAL.

188. By multiplication we have

$\left(a+b\right)^{2}=\left(a+b\right)\left(a+b\right)=a^{2}+2ab+b^{2}$ . .(1)

$\left(a-b\right)^{2}=\left(a-b\right)\left(a-b\right)=a^{2}-2ab+b^{2}$ . .(2)

Rule I. The square of the sum of two quantities is equal to the sum of their squares increased by twice their product.