10 F1.02: Examples 4-5
Example 4. Negative numbers and the operation of subtraction.
- [latex]8-6[/latex]
- [latex]-6+8[/latex]
Solution: When we do subtraction problems by hand or work with negative numbers, we usually write the negative sign and the subtraction symbol in exactly the same way, so we think of them as the same. But calculators treat them differently. Use your calculator to find [latex]8-6[/latex].
Now try to find [latex]-6+8[/latex]. Using the subtraction symbol won’t work here. The key you need is probably labeled [latex]+/-\,\,\,\,\,or\,\,\,\left(-\right)[/latex]. Find that key and experiment with it until you see how to use your calculator to find [latex]-6+8[/latex].
Most students go through entire algebra courses and never need the negative number key because they handle all the sign parts of the problem mentally. For example, we’d just say [latex]-6+8=8-6[/latex] and then use the calculator for this resulting problem. But when we learn about trigonometry later in the course, we will need to be able to fully handle negative numbers, so we’ll need to use this key.
Discussion: Order of operations.
There are three types of mathematical expressions which we write, by hand, without parentheses, but which need parentheses when entering them into a calculator or spreadsheet.
| Evaluate each expression | Enter this | Why? |
| [latex]{{3}^{2x}}[/latex], where [latex]x=3[/latex] |
3 ^ (2*3) = or [latex]3\,\,{{y}^{x}}(2*3)=[/latex] |
In the original expression, the placement of the symbols indicates that the exponent is 2 times x. But when we have to just enter symbols one after the other – on the same line – we have to use parentheses to clarify what is in the exponent.
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| [latex]\frac{{{x}^{2}}-6x}{4x+2}[/latex], where [latex]x=3[/latex] |
(3^2-6*3)/(4*3+2) = or (32-6*3)/(4*3+2) = |
In the original expression, the placement of the symbols indicates that the entire numerator is divided by the entire denominator. But when we have to just enter symbols one after the other – on the same line – we have to use parentheses to clarify what is to be divided by what.
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| [latex]\sqrt{4x+13}[/latex], where |
(4*3+13) =
OR 4*3+13 = [latex]\sqrt{{}}[/latex] = |
In the original expression, the fact that the expression was completely under the square root symbol made it clear. In the calculator, we have to use parentheses to say that.
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Example 5. For each of the expressions above, evaluate it by hand and then evaluate it with your calculator. By hand
| [latex]{{3}^{2x}}[/latex], where [latex]x=3[/latex] | [latex]\begin{align}&\,\,\,\,{{3}^{2\cdot3}}={{3}^{6}}=3\cdot3\cdot3\cdot3\cdot3\cdot3\\&=9\cdot3\cdot3\cdot3\cdot3=27\cdot3\cdot3\cdot3=81\cdot3\cdot3=243\cdot3=729\\\end{align}[/latex] |
| [latex]\frac{{{x}^{2}}-6x}{4x+2}[/latex], where [latex]x=3[/latex] |
[latex]\frac{{{3}^{2}}-6\cdot3}{4\cdot3+2}=\frac{9-18}{12+2}=\frac{-9}{14}[/latex] Now use your calculator to find this is –0.642857 |
| [latex]\sqrt{4x+13}[/latex], where [latex]x=3[/latex] | [latex]\sqrt{4x+13}=\sqrt{4\cdot3+13}=\sqrt{25}=5[/latex] |
Plug these into your calculator using the expressions above, with parentheses
| Evaluate each expression | Enter this |
| [latex]{{3}^{2x}}[/latex], where [latex]x=3[/latex] |
3 ^ (2*3) = 729 OR [latex]3\,\,{{y}^{x}}(2*3)=[/latex] 729
|
| [latex]\frac{{{x}^{2}}-6x}{4x+2}[/latex], where [latex]x=3[/latex] |
(3^2-6*3)/(4*3+2) = –0.642857
OR (32-6*3)/(4*3+2) = –0.642857
|
|
[latex]\sqrt{4x+13}[/latex], where [latex]x=3[/latex] |
[latex]\sqrt{{}}[/latex](4*3+13) = 5
OR 4*3+13 = [latex]\sqrt{{}}[/latex] = 5
|