6. Free a (a+b) and (a−b) √a+b of coefficients.

7. Reduce x2, y3, and 4 to equivalent expressions having

8. Free x (y) and a(z) of coefficients.

9. Free x6 (y)3 and 16 (x) of coefficients.

10. Show that 2+1+2
2x8 (1 + 2 *) *= (x2 + y2) 3

4. Reduction to same degree.

Reduce √3, 2, and 2 to the same degree.

Note.—The operation may be shortened by remembering that both index and exponent may be multiplied by the same number [P. 89].

Reduce Va2, a, and Va1 to the same degree.

Note.-The common index is the L. C. M. of the given indices.

Solution: 48 = 4√4x2 = 4 × √4 × √2

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2. Find the value of 2-81 a++8√3a2-2 a V-24 a.

Solution :

2/81 a* = 2 3/ — 27 a3 × 3 a = 2 × 3/

· 27 a3× √3a=

=- 6a3a = +83 a3 × 3 a = +8 × √√√ a3 × √3a =+8a/3a

+ 8 = 3 a+

-

-2a/24a=-2a-8×3a=-2a ×

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15. (a3 +2 a2 b + a b2) § − (a3 − 2 a2 b + a b2) $

16. √a3 — 2 a2 b + a b2 ± √ a3 + 2 a2 b + a b3

19. (3 √10 + 4 √5 – 6 √15) ÷ √5

20. (5 a √ a2 — b2 — 10 a b √ a + b) ÷ 5 a √ a + b

21. (x+2√xy + y) ÷ (√x + √y)

22. (x2+xy+y3) ÷ (x + √ xy+y)

Involution of Radicals.

Illustrations.-1. Raise 5 to the second power.
Solution: (5) = (5})2 = 5} [P. 99] = 3/52 = 3/25.
2. Raise a to the third power.

Solution: (√√√a)3 = (at)3 = a* [P. 99] = a1 [P. 89] = √√/a.

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3. Raise a to the sixth power.

Solution: (a) = (a})6 = a} [P. 99] = a2 [P. 89].