Example 4. Divide 8108 by 26. 8/108÷2√6=4/18=4√2 × √9=3×4√2=12√2. Example 5. Divide (a2b3d3) by dł. (a2b2d3)o' ÷ d2 = (a2b2 d3 jo ÷ (d3 }& = (a2 b2 )'3 = (ab)3. Example 6. Divide (16a3 — 12a2x) by 2a. First raise 2a to the second power, which=4a2: Then, (16a3 — 12a3x)13 ÷ 4a2 =(4a—3x)13. INVOLUTION OF RADICAL QUANTITIES. Example 2. a-√b a a2-a√b -a√b+b a2-2a√b+b=Square of a- √b. a-√b a3-2a3 √b+ab ~a3 √b+2ab-b√b a3-3a2 √b+3ab-b√b=Cube. Example 3. 2d+√x 2d+ √x 4d2 +2d√x +2d√x+x 4d2+4d √x+x=Square of 2d + √г. 2d+√x 8d3 +8d3 √x+2dx +4d2 √x+4dx+xx 8d3+12d3 √x+6dx+x√x=Cube. Article 293. b.) √10-√2-3 is here multiplied into ✓10-√2−√3 itself, and the result appears on the next 10+√2+√3) page. ✓100 * -√4-√6-9-5-2/6 the product of ✓10-√2√3 into 10-2-√3. 5-2/6 5+2√6 25-10√6 25 +106-436 −4√✓/36=25—24=1= the product of. 5-26 into 5+2 √6. The numerator or denominator of a fraction may be cleared from radical signs without altering the value of the frac tion, if the numerator and denominator be both multiplied by a factor which will render either of them rational. See Day's Algebra, Art. 293, d. Example 8. The fraction 8 = 4 8x(√3-√2-1) (−√2) 2√+2√2 ̄(√3+√2+1) (√3−√2—1) (— √2) −2√6+2√2. For the first two factors in the denominator, √3+√2+1, and √3−√2-1, when multiplied together-22, which multiplied into the remaining factor, - √2, becomes 4. The factors of the numerator when mul. tiplied together=-8√6+√/8√4+82. Therefore the fraction becomes 8√6+8√4+8√2_−2√+2√4+2√2 4 −2√6+2√4+2√4=4−2√6+2√2. 1 = Example 10. a-√b_(a−√b) x (a+√b) _a2 —b = a+ √b (a+√b) (a-√b) a2-b EXAMPLES FOR PRACTICE. Example 1. Find the 4th root of 81a3. //81a3=3a=3a3a=the 4th root of 81a2. Example 2. Find the 6th root of (a+b) ̃3. =1 (a+b)-3=(a+b)-&=3⁄4/(a+1) ̃ ̄3=(a+b)-—6th root of (a+b)-3. Example 3. Find the nth root of (x—y)ś. (x—y)t×n =(x—y)on = (x-9)=the nth root of (a—b). Example 4. Find the cube root of -125a3x. -5ax2 Example 7. Find the square root of x2 -6bx+9b2. x2-6x+963 being the square of a binomial, the square root of it according to Art. 265-x-3b. Example 8. Find the square root of a +ay+ ya 4 By Art. 265, the square root of a2 +ay+? y3 2 Example 9. Reduce ax2 to the form of the 6th root. Example 10. Reduce -3y to the form of the cube root. (-27y3)= 2/27y3=Answer. Example 11. Reduce a2 and a3 to a common index. The indices are 2 and, which reduced to a common index and . (a) and a=Answer. Example 12. Reduce and 5+ to a common index. (44) 11⁄2 and (53) 11⁄2=(256) '1⁄2 and (125)î'1⁄2=Answer. Example 13. Reduce a1 and to the common index §. (a) and (b2)=Answer. Example 14. Reduce 2a and 44 to the common index f. (24) and (42)§=168, and 168=Answer. Example 15. Remove a factor from √294. √294=√49× √6=7√6. Example 16. Remove a factor from √x3-a2x2. √x2 × √x-a2=x√x—a2. Example 17. Find the sum and difference of 16a2x and 4a2x. √16a2x=√16a2 × √x=4a√x. Example 18. Find the sum and difference of 3/192 and 3/24 /192/64×3/3=43/3. √24= 8×33/3=23/3. 4 3/3+23/3-6 3/3=sum. 43/3-23/3=23/3=difference. Example 19. Multiply 73/18 into 53/4. 7/18×53/4=353/72=703/9. Example 20. Multiply 4+2√2 into 2-√2. 4+2√2×2-2=8-24-8-4=4. Example 21. Multiply a(a+√c)a into b(a−√e)*. a(a+√c)3×b(a−√e)§=ab(a2—c)§. Example 22. Multiply 2(a+b) into 3(a+b)". 2(a+b)="×3(a+b)==6(a+b). n m Example 23. Divide 654 by 3√2. 6/54÷3√2=2√27=6√3. Example 24. Divide 43/72 by 23/18.. 43/72÷23/18=23/4. Example 25. Divide ✓7 by 3/7. √7÷2/7=√7 Example 26. Divide 83/512 by 43/2. Example 27. Find the cube of 1721. Example 28. Find the square of 5+√2. Example 29. Find the 4th power of √6. Example 30. Find the cube of √x−√b. (x2√bx+b)x(√x−√b) = x√x − 2√bx2 + b√x -x√b+2√b2x-bb. Example 31. Find a factor which will make y rational. Vy×V/y2=y; or, y3×3—y. Example 32. Find a factor which which will make √5— ✓ rational. (√5−√x)× (√5+√x)=5−x, |