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6a2 (b+x)2+4a(b+x)3+(b+x)=a1 +4a3 (b+x) + 6a3 (b2+2bx+x2)+4a(b3+3b2x+3bx2 + x3) + (b1 + 4b3x + 6b2x2+4b3x+4)=the answer.

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efficients. 1-y+y2—y3 +y1 the terms.

2.4.6 2.4.6

4 2.4.6.8

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the indices. 1§.—1—32.+174 22-17} -1 ༔ ༤

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23 .1

x3

8

3 3

=

-the terms.

X

2.5

2.5.8

+

3.6.9.12

the co-efficients. hx3×

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2.y 3.6.a5

EVOLUTION OF COMPOUND QUANTITIES.

Example 3. What is the 5th root of a5 +5a4b+10a3b2+ 10a2b3+5ab4 +65 ?

a5+5a1b+10a3b2 +10a2b3+5aba +b5(a+b

a5

5a4)

5a4b+10a3b2+10a2b3+5ab4+b5 a5+5a+b+10a3b2+10a2b3+5ab4+b5

Example 4. What is the cube root of a3-6a2b+12ab3

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3a2) -6a2b+12ab2-8b3

a3-6a2b+12ab2-853

Example 3. Art. 485. What is the square root of ao —2a5 +3a4-2a3 +a2 ?

a62a5+3a1-2a3 +a2 (a3—a2+a

ασ

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Example 4. What is the square root of a++4a2b+4b2 4a2-8b+4?

a2+4a2b+4b2-4a2 -8b+4(a2+2b—2

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Six examples under article 486.

Example 1. Find the square root of x4-4x+6x2-4x+1. x4-4x+6x-4x+1(x2-2x+1

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Example 2. Find the cube root of x6—6x5+15x4—20x3+ 15x2-6x+1.

x-6x515x4-20x3+15x2-6x+1(x2-2x+1

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Example 3. Find the square root of 4x4-4x3+13x2-6x+9.

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Example 4. Find the 4th root of 16a4-96a3x+216a2x2.

216ax3 +81x1.

-

16a1-96a3x+216a2x2-216ax3 +81x4, the quotient

16a4

[=√14a2-12ax+6x=2a-3x

8a2-12ax)-96a3x+216a2x2

-96a3x+144a2x2

8a2-24ax+9x2)72a2x2 -216ax3 +81x1

72a2x2-216ax3 +81x4

Example 5. Find the 5th root of x5 +5x4+10x3+10x2+

5 +1.

5x4)

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5x4+10x3+10x2+5x+1

x5+5x4+10x3 +10x2+5x+1

Example 6. Find the 6th root of a6-6a5b+15a4 b2 — 20a3b3+15a2ba —бab5 +b®.

a6-6a5b-15a4b2-20a3b3+15a2b1-6ab5+b® (a−b

a6

6a5)-6a5b+15a4b2-20a3b3+15a2b4-6ab5+b6 -6a5b+15a+b2 -20a3b3 +15a2b1—6ab5 +b¤

ROOTS OF BINOMIAL SURDS. Art. 486. c.

Example 2. Find the square root of 11+6√2. a=11. a2=121. √b=6√2. b=36×2.

b=72.

√ a2 -b = √49-7-the difference. ✓ 11+6√2 =

11+7

+

11-7

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=√9+√2=3+√2.

Example 3. Find the square root of 6-2√5.

a=6.

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a2=36. √b=2√5. b=4×5=20.

= √3 36-20 = √16 = 4 the difference.

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Va2-b √6-2√5

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Example 4. Find the square root of 7+4√3. a=7. a2=49. √b=4√3. b=16×3=48. =√49-48=1=the difference. √7+√3=

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b=4×10=40.

√a2-b=√49-40-/9-3-the difference. √7-210

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Under INFINITE SERIES are included indeterminate co. efficients, summation of series, recurring series, and method of differences. Under the heads infinite series and indeterminate co-efficients, are four methods for expanding an expression into an infinite series. Summation of series is find. ing the sum of the terms. Recurring series is when a certain number of contiguous terms are so related to each other, that any of the following may be found by recurring to those which precede. Method of differences is finding the value of a limited number of terms, which depends on finding the several orders of differences belonging to the series.

1+a

Example 4. Art. 488. Reduce to an infinite series.

1-a

1-a)1+a(1+2a+2a2+2a3+2a1+2a5 +2a +2a7 &c. 1-a 1st subtrahend.

2a

2a-2a2 = 2d subtrahend.

2a2

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