Do Neutral Operations with Exponents

You will learn how to algebraically solve a^n + b^n = a^n * b^n =c^n (whereby two operators are held neutral to each other between constants, rendering a "tipping point") and to graph the final a^n and b^n as x and y1 with c as y2. This article has bearing on the Pythagorean Theorem, among other formulas, e.g. in Allometry where y=kx^a. You will also learn how to solve and make a table for a^n - b^n = a^n / b^n =c^n, which has applicability in Nature, especially to the conservation principles and spaces around organs and organism parts, including Special Relativity, E = mc^2.

Steps

  • Become familiar with the chart-image to create:

a^n + b^n = a^n * b^n = c^n

  1. Open a new workbook in Excel from the desktop, from the dock, or from within your Applications folder inside the Microsoft folder. Double click on Excel (either the green X on the dock or the app title in the folder) and select File New Workbook.
  2. In Preferences, set R1C1 to unchecked or Off, set Ribbon to checked or On, set Show Formula Bar to checked or On and Set Calculate to Automatically.
  3. Click in the far upper left top corner above the 1 of row 1 and to the left of column A. Doing so will select the entire worksheet. Format Cells Alignment Center. Format column width 1.5".
  4. Title the first worksheet, "Data" and title the second worksheet "Saved Charts"; save the workbook as "Do NeuOps w Exponents" into an appropriate folder such as 'Microsoft Excel Imagery', 'Neu Ops' or 'wikiHow Articles'.
  5. Enter the Row 1 Headings:
    • Enter to cell A1, a
    • Enter to cell B1, b
    • Enter to cell C1, n
    • Enter to cell D1, a^n
    • Enter to cell E1, b^n
    • Enter to cell F1, (a^n -1)
    • Enter to cell G1, a^n /(a^n - 1) = b^n (see the previous article if you don't know how this was arrived at)
    • Enter to cell H1, a^n + b^n = c^n
    • Enter to cell I1, a^n * b^n = c^n
    • Enter to cell J1, c
    • Enter to cell K1, c^n check figure
  6. Enter the preliminary input values and formulas
    • Enter to cell A2, the preliminary input value, 10, and Format Cells Fill yellow
    • Edit Copy cell A2 and Edit Paste it to cell A3 and enter the formula in A3, =A2-1
    • Edit Copy cell A3 and Edit Paste it to cell range A4:A10
    • Select cell A2 and Format Cells Border Red Boldest Outline (for an input cell)
    • Edit copy cell A2 and Edit Paste it to cell C2; enter the value 2 to cell C2.
    • Enter to cell C3 the formula, =C2, and Edit Copy it and Edit Paste it to cell range C4:C10
    • Select columns A:F and Insert Name Create in Top Row, OK.
    • Enter to cell B2 the formula, =b_n^(1/n)
    • Edit Copy cell B2 and Edit Paste it to cell range B3:B10
    • Enter to cell D2 the formula, =a^n
    • Enter to cell E2 the formula, =G2
    • Enter to cell F2 the formula, =a^n-1
    • Enter to cell G2 the formula, =a_n/a_n___1 (by using Insert Name Paste)
    • Enter to cell H2 the formula, =a_n+b_n
    • Enter to cell I2 the formula, =a_n*b_n
    • Enter to cell J2 the formula, =I2^(1/n)
    • Select column J and Insert Name Create Name in Top Row, OK.
    • Enter to cell K2 the formula, =c_^n
    • Edit Copy cell range D2:K2 to cell range D3:K10
    • Match the results in columns H and I as proof of Operator Neutrality.
    • Select Row 1 and Format Cells Font Underline, Bold.

Chart the Results

  1. Enter the headers and data:
    • Enter to cell C13 the header, x = a^n
    • Enter to cell D13 the header, y1 = b^n
    • Enter to cell E13 the header, y2 = c
    • Select cell range C13:E13 and Format Cells Font Underline, Bold.
    • Enter to cell C14 the formula, =D2
    • Enter to cell D14 the formula, =E2
    • Enter to cell E14 the formula, =J2
    • Select cell range C14:E14 and Edit Copy it and Edit Paste it to cell range C15:E22.
  2. Create the chart
    • Select cell range C14:E22 and either Insert Chart or do Chart Wizard or do the Ribbon, Charts, All, Scatter, Smooth Lined Scatter. A new chart will appear atop your data. Move its top left corner to the inside of cell F14 and grab its lower right corner when the cursor is over it and turns into a double headed arrow, then expand it to cell K40 by dragging down and to the right.
    • Double click in the Plot Area and set the fill to sky blue. Double click in the Chart Area and set the fill to medium purple.
    • Double click on the Series 1 at the bottom and edit the series formula in the formula bar until it reads, =SERIES("{x=a^n, y1=b^n}",Data!$C$14:$C$22,Data!$D$14:$D$22,1).
    • Double click on the Series 2 at the top and edit the series formula in the formula bar until it reads, =SERIES("{x=a^n, y2=c}",Data!$C$14:$C$22,Data!$E$14:$E$22,2).
    • Select from the Ribbon Chart Layout, select Chart Title and edit the Chart Title to be centered, top and read as follows: "(a^n = x [+ and *] b^n = y1) = c^n; c = y2
    • for a = [10..2] and n=2" (that is, there is a return before the "for")
    • Delete any other series that have been inadvertently created. Your chart should resemble this one.

a^n - b^n = a^n / b^n = c^n

  1. Select cell range A26:A30 and Format Cells Alignment Left, Font Bold.
  2. Enter the algebraic solution the isolates and defines a^n in terms of b^n and 1:
    • Enter to cell A26 the equation, "a^n - b^n = a^n / b^n = c^n", w/o quotes;
    • Enter to cell A27 the equation, "a^n -a^n - b^n = a^n / b^n -a^n";
    • Enter to cell A28 the equation, " - b^n = a^n*(1/(b^n) - 1)" w/ leading space;
    • Enter to cell A29 the equation, " - b^n/(1/(b^n) - 1) = a^n" w/ leading space;
    • Enter to cell A30 the equation, "b^n/(1 - 1/(b^n)) = a^n" (by multiplying the left side by -1/-1 and rearranging the denominator).
  3. Set up the table for a^n - b^n = a^n / b^n = c^n:
    • Edit Copy cell range A1:K1 and Edit Paste it to cell A42;
    • Edit the formula in cell F42 in the formula bar to: (1 - 1/(b^n))
    • Edit the formula in cell G42 in the formula bar to: b^n /(1 - (b^n)) = a^n
    • Edit the formula in cell H42 in the formula bar to: a^n - b^n = c^n
    • Edit the formula in cell I42 in the formula bar to: a^n / b^n = c^n
    • Edit Copy cell range A2:A10 and Edit Paste it to cell B43;
    • Edit Copy cell range C2:C10 and Edit Paste it to cell C43;
    • Enter to cell E43 the formula, =B43^C43, and select cell range E43:E51 and Edit Fill Down.
    • Enter to cell F43 the formula, =(1-1/(E43)), and select cell range F43:F51 and Edit Fill Down.
    • Enter to cell G43 the formula, =E43/F43, and select cell range G43:G51 and Edit Fill Down.
    • Enter to cell D43 the formula, =G43, and select cell range D43:D51 and Edit Fill Down.
    • Enter to cell A43 the formula, =D43^(1/C43), and select cell range A43:A51 and Edit Fill Down.
    • Enter to cell H43 the formula, =D43-E43, and select cell range H43:H51 and Edit Fill Down.
    • Enter to cell I43 the formula, =D43/E43, and select cell range I43:I51 and Edit Fill Down.
    • Enter to cell I43 the formula, =D43/E43, and select cell range I43:I51 and Edit Fill Down.
    • Enter to cell J43 the formula, =I43^(1/C43), and select cell range J43:J51 and Edit Fill Down.
    • Enter to cell K43 the formula, =J43^C43, and select cell range K43:K51 and Edit Fill Down.
    • Note that cell range H43:H51 = I43:I51 and that the Neutral Operation equation of a^n - b^n = a^n / b^n = c^n is true, in that the values in J43:J51 for c exist. That a difference equals a ratio or proportion is important because, in Nature, certain ratios or proportions are necessary and they may not always be constructed additively. An example of this is the space in the mouth or the spaces between teeth -- such a space is essential for the animal's survival but may not be created additively, only by making sure not too much is added in fact, i.e that a proportion is maintained as a difference. See Allometry and Spherical Cap in Wikipedia[1][2] for information on how body mass varies and on the availability of united spherical or hyperspherical surfaces to reactants/solvents -- how these may be calculated and/or compared to expected values. Adding a coefficient to the above neutral operation is easily done.
  4. See also in the citations below the case for the Conservation of Energy, with respect to the Theory of Special Relativity, E = mc^2.[3]

Helpful Guidance

  1. Make use of helper articles when proceeding through this tutorial:
    • See the article How to Do the Sub Steps of Neutral Operations for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
    • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

Related Articles

Sources and Citations

  1. ↑ https://en.wikipedia.org/wiki/Allometry
  2. ↑ https://en.wikipedia.org/wiki/Spherical_cap
  3. ↑ https://en.wikipedia.org/wiki/Conservation_of_energy
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