#set document(title: "12.2 Entering Expressions", author: "OpenStax / XYZ Homework")
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== 12.2#h(0.6em)Entering Expressions


    
      
=== Parentheses


      #strong[Order of Operations]: The calculator follows the standard order of operations.


      #examplebox("Example 1")[][
Compute #math.equation(block: false, alt: "2 plus 3 times 4")[$2 + 3 ⋅ 4$].


        Enter #math.equation(block: false, alt: "2")[$#h(2em) 2$] +  #math.equation(block: false, alt: "3")[$3$]  #math.equation(block: false, alt: "4")[$4$]


        Ans. #math.equation(block: false, alt: "14")[$14$]
]


      #examplebox("Example 2")[][
Compute #math.equation(block: false, alt: "open parenthesis 2 plus 3 close parenthesis times 4")[$( 2 + 3 ) ⋅ 4$].


        Enter $#h(2em)$ (  #math.equation(block: false, alt: "2")[$2$]  +  #math.equation(block: false, alt: "3")[$3$]  )    #math.equation(block: false, alt: "4")[$4$]


        Ans. #math.equation(block: false, alt: "20")[$20$]
]


    
    
      
=== Fractions


      #examplebox("Example 3")[][
Compute #math.equation(block: false, alt: "the fraction 1 plus 3 over 2")[$display(frac(1 + 3, 2))$].


        Enter $#h(2em)$ (  #math.equation(block: false, alt: "1")[$1$]  +  #math.equation(block: false, alt: "3")[$3$]  )    #math.equation(block: false, alt: "2")[$2$]


        Ans. #math.equation(block: false, alt: "2")[$2$]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
GeoGebra displays "built-up" fractions like #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$]. Once we enter the numerator and select the  key, the cursor is in the denominator and will stay there until we use an arrow key on the keyboard to move the cursor outside the fraction, or until we press . The arrow keys on the virtual keyboard are shown in the keyboard image.
]


      #examplebox("Example 4")[][
Compute #math.equation(block: false, alt: "the fraction 1 over 2 times 3")[$display(frac(1, 2 ⋅ 3))$].


        Enter #math.equation(block: false, alt: "1")[$#h(2em) 1$]  #math.equation(block: false, alt: "2")[$2$]  #math.equation(block: false, alt: "3")[$3$]


        Ans. #math.equation(block: false, alt: "the fraction 1 over 6")[$display(frac(1, 6))$]


        In the output history, there is an approximation icon that looks like the "#math.equation(block: false, alt: "approximately equals")[$≈$]" symbol in a blue square. To get a decimal approximation of the fraction, click on that icon.


        Ans. #math.equation(block: false, alt: "0.1666666666667")[$0.1666666666667$]
]


      #examplebox("Example 5")[][
Compute #math.equation(block: false, alt: "the fraction 1 over 2 times 3")[$display(frac(1, 2)) ⋅ 3$].


        Enter #math.equation(block: false, alt: "1")[$#h(2em) 1$]  #math.equation(block: false, alt: "2")[$2$]   #math.equation(block: false, alt: "3")[$3$]


        Ans. #math.equation(block: false, alt: "the fraction 3 over 2")[$display(frac(3, 2))$]
]


    
    
      
=== Exponents and Powers


      The key for exponents and the key for squaring are shown in the keyboard image.


      #examplebox("Example 6")[][
Evaluate #math.equation(block: false, alt: "57 squared")[$57^(2)$].


        Enter #math.equation(block: false, alt: "57")[$#h(1em) 57$], then the squaring key, then


        OR


        Enter #math.equation(block: false, alt: "57")[$#h(1em) 57$], then the exponent key, then #math.equation(block: false, alt: "2")[$2 " "$]


        Ans. #math.equation(block: false, alt: "3249")[$3249$]
]


      #examplebox("Example 7")[][
Evaluate #math.equation(block: false, alt: "2 to the power 10")[$2^(10)$].


        Enter #math.equation(block: false, alt: "2")[$#h(1em) 2$], and then the exponent key, 10


        Ans. #math.equation(block: false, alt: "1024")[$1024$]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
GeoGebra nicely displays powers such as #math.equation(block: false, alt: "10 to the power 7")[$10^(7)$], with the exponent raised like a superscript. Once we enter the base and select the exponent key, the cursor is in the exponent and will stay there until we use the arrow keys to move the cursor outside the power, or until we press . The arrow keys on the virtual keyboard are shown in the keyboard image.
]


      #examplebox("Example 8")[][
Evaluate #math.equation(block: false, alt: "8 to the power 2 / 3")[$8^(2 / 3)$].


        Enter #math.equation(block: false, alt: "8")[$#h(1em) 8$], then the exponent key, then #math.equation(block: false, alt: "2")[$2$]  #math.equation(block: false, alt: "3")[$3$]


        Ans. #math.equation(block: false, alt: "4")[$4$]
]


      #examplebox("Example 9")[][
Evaluate #math.equation(block: false, alt: "the fraction 8 squared over 3")[$display(frac(8^(2), 3))$].


        Enter #math.equation(block: false, alt: "8")[$#h(1em) 8$], then the exponent key, then #math.equation(block: false, alt: "2")[$2$]   #math.equation(block: false, alt: "3")[$3$]


        Ans. #math.equation(block: false, alt: "the fraction 64 over 3")[$display(frac(64, 3))$]
]


    
    
      
=== Square Roots


      The key for square roots is shown in the keyboard image.


      #examplebox("Example 10")[][
Evaluate #math.equation(block: false, alt: "the square root of 2")[$sqrt(2)$].


        Select the square root key, then enter #math.equation(block: false, alt: "2")[$" " 2$]


        Ans. #math.equation(block: false, alt: "1.414213562")[$1.414213562$]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
GeoGebra nicely displays square roots such as #math.equation(block: false, alt: "the square root of 9 plus 16")[$sqrt(9 + 16)$], with the radicand inside the square root "house". Once we select the square root key, the cursor is in under the radical and will stay there until we use the arrow keys to move the cursor outside, or until we press . The arrow keys on the virtual keyboard are shown in the keyboard image.
]


      #examplebox("Example 11")[][
Evaluate #math.equation(block: false, alt: "the square root of 9 plus 16")[$sqrt(9 + 16)$].


        Select the square root key, then enter #math.equation(block: false, alt: "9")[$" " 9 " "$]  #math.equation(block: false, alt: "16")[$" " 16 " "$]


        Ans. #math.equation(block: false, alt: "5")[$5$]
]


      #examplebox("Example 12")[][
Evaluate #math.equation(block: false, alt: "the square root of 9 plus 16")[$sqrt(9) + 16$].


        Select the square root key, then enter #math.equation(block: false, alt: "9")[$" " 9 " "$]   #math.equation(block: false, alt: "16")[$" " 16 " "$]


        Ans. #math.equation(block: false, alt: "19")[$19$]
]


    
    
      
=== Other Roots


      If we click on the "f(x)" on the keyboard in the Algebra View, we see other built-in keys.


      #figure(figph[GeoGebra calculator f(x) keyboard], alt: "GeoGebra calculator f(x) keyboard", caption: [GeoGebra calculator f(x) keyboard])


      #examplebox("Example 13")[][
Compute #math.equation(block: false, alt: "the cube root of 1728")[$root(3, 1728)$].


        Choose the f(x) keyboard, select the nth root key, go back to the standard 123 keyboard, then enter #math.equation(block: false, alt: "3")[$" " 3 " "$]  #math.equation(block: false, alt: "1728")[$" " 1728 " "$]


        Ans. #math.equation(block: false, alt: "12")[$12$]
]


      #examplebox("Example 14")[][
Compute #math.equation(block: false, alt: "the 10th root of 2 times 512")[$root(10, 2 ⋅ 512)$].


        Choose the f(x) keyboard, select the nth root key, go back to the standard 123 keyboard, then enter #math.equation(block: false, alt: "10")[$" " 10 " "$]  #math.equation(block: false, alt: "2")[$" " 2 " "$]  #math.equation(block: false, alt: "512")[$" " 512 " "$]


        Ans. #math.equation(block: false, alt: "2")[$2$]
]


    
    
      
=== Absolute Value


      GeoGebra nicely displays the vertical bars of the standard absolute value notation, such as in the expression #math.equation(block: false, alt: "vertical bar 21 times 54 minus 81 vertical bar")[$| 21 ⋅ 54 − 81 |$]. Once we select the absolute value key, the cursor is in between the vertical bars and will stay there until we use the arrow keys to move the cursor outside, or until we press . The arrow keys on the virtual keyboard are shown in the keyboard image.


      #examplebox("Example 15")[][
Evaluate #math.equation(block: false, alt: "the fraction vertical bar 21 times 54 minus 81 vertical bar over minus 9")[$display(frac(| 21 ⋅ 54 − 81 |, − 9))$].


        Select the absolute value key, then enter #math.equation(block: false, alt: "21")[$" " 21 " "$]  #math.equation(block: false, alt: "54")[$" " 54 " "$]  #math.equation(block: false, alt: "81")[$" " 81 " "$]    #math.equation(block: false, alt: "minus 9")[$" " − 9 " "$]


        Ans. #math.equation(block: false, alt: "minus 117")[$− 117$]
]


    
    
      
=== Scientific Notation


      The GeoGebra calculator does not display scientific notation by default. The calculator can display scientific notation in the Grahics View by using a ScientificText command.


      #examplebox("Example 16")[][
Compute #math.equation(block: false, alt: "123 , 456 , 789 squared")[$123 , 456 , 789^(2)$].


        Enter #math.equation(block: false, alt: "123456789")[$123456789$]


        Ans. #math.equation(block: false, alt: "15241578750190522")[$15241578750190522$]


        We will enter ScientificText in the input box.


        - On the computer, typing the first three letters (sci) is enough for the full command to appear as an option.
- On a phone app, we tap on the overflow menu (three horizontal dots) on the top right of the virtual keyboard (see keyboard image), tap in the Search in All Commands field, then start to input ScientificText until the full command appears as an option.


        We click on ScientificText, and the cursor is appropriately positioned, ready for us to enter an expression. We can either repeat the calculation we entered above, or, more efficiently, select the last answer key ans and .


        The result #math.equation(block: false, alt: "1.52415787501905 times 10 to the power 16")[$1.52415787501905 × 10^(16)$], appears on the grid in the Graphics View. We can drag the text within the Graphics View, change its color, and/or delete it.


        We could also type in the entire command scientifictext using the ABC keyboard, then use the 123 keyboard to enter the parentheses and numbers.
]


    
    
      
=== Editing an Entry


      We can edit an expression without starting again. The backspace key will remove the character to the left of the cursor. We can move the cursor within an input line using the arrow keys or by clicking in the appropriate place.


      We can recall any previous entry by finding it in the output history (scrolling up if necessary), tapping on the kebab menu icon (three vertical dots), and selecting Duplicate. The command will be copied into the current input box, which we can edit before pressing .


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[If the menu does not appear]


        Sometimes when using the GeoGebra app embedded in this textbook, the menu does not appear when we tap on the kebab menu icon. This may occur when we are at a place in the section far from the top of the webpage. Try scrolling within the webpage to the top, and tap on the kebab menu icon again.
]


      If we just want to use the most recent result in the current command, we use the ans key.


      #examplebox("Example 17")[][
Evaluate #math.equation(block: false, alt: "5 squared plus 12 squared")[$5^(2) + 12^(2)$] and then take the square root of the sum.


        Enter #math.equation(block: false, alt: "5")[$5$], then the square key, then  #math.equation(block: false, alt: "12")[$12$], the square key again, then


        Ans. #math.equation(block: false, alt: "169")[$169$]


        Now select the square root key, then ans


        Ans. #math.equation(block: false, alt: "13")[$13$]
]