Square and Cube Roots

Excerpts

Square and Cube Roots
Since the square of $a + b$ is $a^{2} + 2ab + b^{2}$, the square root of $a^{2} + 2ab + b^{2}$ is $a + b$. It is required to find a method of extracting the root $a + b$ when $a^{2} + 2ab + b^{2}$ is given.
Square and Cube Roots
The same method will apply to longer expressions, if care be taken to obtain the *trial-divisor* at each stage of the process, *by doubling the part of the root already found*, and to obtain the *complete divisor by annexing the new term of the root to the trial-divisor*.
Square and Cube Roots
Since the cube of $a + b$ is $a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$, the cube root of a$^{3} + 3a^{2}b + 3ab^{2} + b^{3}$ is $a + b$.
Square and Cube Roots
If, therefore, an integral square number is divided into groups of two figures each, from the right to the left, the number of figures in the root will be equal to the number of groups of figures.
Square and Cube Roots
If the square root of a number has decimal places, the number itself will have *twice* as many.
Square and Cube Roots
If a number contain an *odd* number of decimal places, or if any number give a *remainder* when as many figures in the root have been obtained as the given number has groups, then its exact square root cannot be found.
Square and Cube Roots
It will be noticed that each successive trial-divisor may be obtained by taking the preceding complete divisor with its *last term doubled*.

Equations

Square and Cube Roots
2ab + b^{2} = (2a + b)b
The remainder left after subtracting a squared from the square of a + b factors as (2a + b) times b, which is why the divisor is completed by adding the new root term b.
Square and Cube Roots
3a^{2}b + 3ab^{2} + b^{3} = (3a^{2} + 3ab + b^{2})b
The remainder left after subtracting a cubed from the cube of a + b factors as (3a^2 + 3ab + b^2) times b, which is why the complete divisor is formed by adding 3ab + b^2 to the trial divisor 3a^2.

Problems

Exercise 78

Exercise 78, problem 1, p. 153
$a^{2} + 2ab + 2ac + b^{2} + 2bc + c^{2}$.
Printed answer:
- $a + b + c$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles a + b + c
Exercise 78, problem 2, p. 153
$x^{4} + 2x^{3} + 3x^{2} + 2x + 1$.
Printed answer:
- $x^{2} + x + 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**2 + x + 1
Exercise 78, problem 3, p. 153
$x^{4} - 4x^{3}y + 6x^{2}y^{2} - 4xy^{3} + y^{4}$.
Printed answer:
- $x^{2} - 2xy + y^{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**2 - 2*x*y + y**2
Exercise 78, problem 4, p. 153
$4a^{4} - 12a^{3}b + 29a^{2}b^{2} - 30ab^{3} + 25b^{4}$.
Printed answer:
- $2a^{2} - 3ab + 5b^{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*a**2 - 3*a*b + 5*b**2
Exercise 78, problem 5, p. 153
$16x^{6} + 24x^{5}y + 9x^{4}y^{2} - 16x^{3}y^{3} - 12x^{2}y^{4} + 4y^{6}$.
Printed answer:
- $4x^{3} + 3x^{2}y - 2y^{3}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 4*x**3 + 3*x**2*y - 2*y**3
Exercise 78, problem 6, p. 153
$4x^{6} - 4x^{4}y^{2} + 12x^{3}y^{3} + x^{2}y^{4} - 6xy^{5} + 9y^{6}$.
Printed answer:
- $2x^{3} - xy^{2} + 3y^{3}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*x**3 - x*y**2 + 3*y**3

Exercise 79

Exercise 79, problem 1, p. 156
$324$.
Printed answer:
- $18$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 18
On the STU-32 (STU, rpn):
```
324 √x
```
Calculator: +18E+0; the book prints 18. Run on the calculator core at firmware 628c96c.
Exercise 79, problem 10, p. 156
$346,921$.
Printed answer:
- $589$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 589
On the STU-32 (STU, rpn):
```
346921 √x
```
Calculator: +589E+0; the book prints 589. Run on the calculator core at firmware 628c96c.
Exercise 79, problem 11, p. 156
$31,371,201$.
Printed answer:
- $5601$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 5601
Exercise 79, problem 12, p. 156
$1,522,756$.
Printed answer:
- $1234$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 1234
On the STU-32 (STU, rpn):
```
1522756 √x
```
Calculator: +1234E+0; the book prints 1234. Run on the calculator core at firmware 628c96c.
Exercise 79, problem 13, p. 156
$2$.
Printed answer:
- $1.4142\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 1.4142
Exercise 79, problem 14, p. 156
$3$.
Printed answer:
- $1.7320\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE 1.7320
Exercise 79, problem 15, p. 156
$5$.
Printed answer:
- $2.2360\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE 2.2360
Exercise 79, problem 16, p. 156
$6$.
Printed answer:
- $2.4494\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE 2.4494
Exercise 79, problem 17, p. 156
$0.5$.
Printed answer:
- $0.7071\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 0.7071
Exercise 79, problem 18, p. 156
$0.9$.
Printed answer:
- $0.9486\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE 0.9486
Exercise 79, problem 19, p. 156
$\frac{2}{3}$.
Printed answer:
- $0.8164\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE 0.8164
Exercise 79, problem 2, p. 156
$441$.
Printed answer:
- $21$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 21
On the STU-32 (STU, rpn):
```
441 √x
```
Calculator: +21E+0; the book prints 21. Run on the calculator core at firmware 628c96c.
Exercise 79, problem 20, p. 156
$\frac{3}{4}$.
Printed answer:
- $0.8660\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 0.8660
Exercise 79, problem 21, p. 156
$\frac{4}{5}$.
Printed answer:
- $0.8944\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 0.8944
Exercise 79, problem 22, p. 156
$\frac{5}{8}$.
Printed answer:
- $0.7905\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE 0.7905
Exercise 79, problem 3, p. 156
$529$.
Printed answer:
- $23$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 23
On the STU-32 (STU, rpn):
```
529 √x
```
Calculator: +23E+0; the book prints 23. Run on the calculator core at firmware 628c96c.
Exercise 79, problem 4, p. 156
$961$.
Printed answer:
- $31$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 31
On the STU-32 (STU, rpn):
```
961 √x
```
Calculator: +31E+0; the book prints 31. Run on the calculator core at firmware 628c96c.
Exercise 79, problem 5, p. 156
$10.24$.
Printed answer:
- $3.2$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 3.2
Exercise 79, problem 6, p. 156
$53.29$.
Printed answer:
- $7.3$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 7.3
On the STU-32 (STU, rpn):
```
53.29 √x
```
Calculator: +73E-1; the book prints 7.3. Run on the calculator core at firmware 628c96c.
Exercise 79, problem 7, p. 156
$53,824$.
Printed answer:
- $232$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 232
Exercise 79, problem 8, p. 156
$616,225$.
Printed answer:
- $785$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 785
On the STU-32 (STU, rpn):
```
616225 √x
```
Calculator: +785E+0; the book prints 785. Run on the calculator core at firmware 628c96c.
Exercise 79, problem 9, p. 156
$1,500,625$.
Printed answer:
- $1225$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 1225
On the STU-32 (STU, rpn):
```
1500625 √x
```
Calculator: +1225E+0; the book prints 1225. Run on the calculator core at firmware 628c96c.

Exercise 80

Exercise 80, problem 1, p. 159
$x^{3} + 3x^{2} y + 3xy^{2} + y^{3}$
Printed answer:
- $x + y$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x + y
Exercise 80, problem 2, p. 159
$8x^{3} - 12x^{2} + 6x - 1$.
Printed answer:
- $2x - 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*x - 1
Exercise 80, problem 3, p. 159
$8x^{3} - 36x^{2} y + 54xy^{2} - 27y^{3}$.
Printed answer:
- $2x - 3y$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*x - 3*y
Exercise 80, problem 4, p. 159
$64a^{3} - 144a^{2} x + 108ax^{2} - 27x^{3}$
Printed answer:
- $4a - 3x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 4*a - 3*x
Exercise 80, problem 5, p. 159
$1 + 3x + 6x^{2} + 7x^{3} + 6x^{4} + 3x^{5} + x^{6}$.
Printed answer:
- $1 + x + x^{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1 + x + x**2
Exercise 80, problem 6, p. 159
$x^{6} - 3x^{5} + 6x^{4} - 7x^{3} + 6x^{2} - 3x + 1$.
Printed answer:
- $x^{2} - x + 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**2 - x + 1

Exercise 81

Exercise 81, problem 1, p. 163
$46,656$.
Printed answer:
- $36$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 36
On the STU-32 (STU, rpn):
```
46656 ENTER 3 1/x yˣ
```
Calculator: +3599999999999999999999999999999999E-32; the book prints 36. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 10, p. 163
$385,828,352$.
Printed answer:
- $728$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 728
On the STU-32 (STU, rpn):
```
385828352 ENTER 3 GOLD ˣ√y
```
Calculator: +7280E-1; the book prints 728. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 11, p. 163
$1879.080904$.
Printed answer:
- $12.34$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 12.34
Exercise 81, problem 12, p. 163
$1838.265625$.
Printed answer:
- $12.25$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 12.25
On the STU-32 (STU, rpn):
```
1838.265625 ENTER 3
```
```
GOLD ˣ√y
```
Calculator: +12250E-3; the book prints 12.25. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 13, p. 163
$0.01$.
Printed answer:
- $0.2154\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 14, p. 163
$0.05$.
Printed answer:
- $0.3684\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 15, p. 163
$0.2$.
Printed answer:
- $0.5848\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 16, p. 163
$4$.
Printed answer:
- $1.5874\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 17, p. 163
$10$.
Printed answer:
- $2.1544\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 18, p. 163
$87$.
Printed answer:
- $4.4310\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 19, p. 163
$2.5$.
Printed answer:
- $1.3572\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 2, p. 163
$42,875$.
Printed answer:
- $35$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 35
Exercise 81, problem 20, p. 163
$2.05$.
Printed answer:
- $1.2703\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 21, p. 163
$3.02$.
Printed answer:
- $1.4454\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 81, problem 22, p. 163
$\frac{2}{3}$.
Printed answer:
- $0.8735\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE
Exercise 81, problem 23, p. 163
$\frac{3}{4}$.
Printed answer:
- $0.9085\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE
Exercise 81, problem 24, p. 163
$\frac{9}{11}$.
Printed answer:
- $0.9352\dots$
verified: the printed answer passed a computed check
How it was checked
- evaluate: PASS-LOOSE
Exercise 81, problem 3, p. 163
$91,125$.
Printed answer:
- $45$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 45
On the STU-32 (STU, rpn):
```
91125 ENTER 3
```
```
GOLD ˣ√y
```
Calculator: +450E-1; the book prints 45. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 4, p. 163
$274,625$.
Printed answer:
- $65$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 65
On the STU-32 (STU, rpn):
```
274625 ENTER 3 GOLD ˣ√y
```
Calculator: +650E-1; the book prints 65. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 5, p. 163
$110,592$.
Printed answer:
- $48$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 48
On the STU-32 (STU, rpn):
```
110592 ENTER 3 GOLD ˣ√y
```
Calculator: +480E-1; the book prints 48. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 6, p. 163
$258,474,853$.
Printed answer:
- $637$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 637
On the STU-32 (STU, rpn):
```
258474853 ENTER 3 1/x yˣ
```
Calculator: +6369999999999999999999999999999996E-31; the book prints 637. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 7, p. 163
$109,215,352$.
Printed answer:
- $478$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 478
On the STU-32 (STU, rpn):
```
109215352 ENTER 3 GOLD ˣ√y
```
Calculator: +4780E-1; the book prints 478. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 8, p. 163
$259,694,072$.
Printed answer:
- $638$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 638
On the STU-32 (STU, rpn):
```
259694072 ENTER 3 GOLD ˣ√y
```
Calculator: +6380E-1; the book prints 638. Run on the calculator core at firmware 628c96c.
Exercise 81, problem 9, p. 163
$127,263,527$.
Printed answer:
- $503$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 503
On the STU-32 (STU, rpn):
```
127263527 ENTER 3 1/x yˣ
```
Calculator: +5029999999999999999999999999999997E-31; the book prints 503. Run on the calculator core at firmware 628c96c.