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How to Do Neutral Operations with Exponents
Three Parts:Helpful guidancea^n + b^n = a^n * b^n = c^nChart the Results
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.
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EditSteps
EditPart 1 of 3: Helpful guidance
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1Make 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 Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category: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.
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EditPart 2 of 3: a^n + b^n = a^n * b^n = c^n
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1Open 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.
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2In 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.
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3Click 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".
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4Title 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'.
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5Enter 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
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6Enter 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.
EditPart 3 of 3: Chart the Results
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1Enter 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.
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2Create 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.
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