# Khan/khan-exercises

Improve hints: Exponents 2; Add subhints!

1 parent d8f1612 commit 85580f5e16aa7a65d392c1cf6d227be952a73441 beneater committed Feb 14, 2012
Showing with 165 additions and 9 deletions.
1. +17 −0 css/khan-exercise.css
2. +130 −9 exercises/exponents_2.html
3. +18 −0 utils/subhints.js
 @@ -12,6 +12,23 @@ var { font-style: normal; } .final_answer{ font-weight:bold; } +div.subhint { + border: 1px solid #aaaaaa; + background: #f9f9f9; + display: none; + -moz-border-radius: 4px; + -webkit-border-radius: 4px; + border-radius: 4px; + margin-left: 20px; + margin-right: 20px; + padding: 10px; +} + +a.show-subhint { + font-size: 12px; + font-style: italic; +} + #workarea { margin-left: 30px; } #hintsarea { margin-left: 50px; } #answer_area ul { list-style: none; }
 @@ -1,13 +1,13 @@ - + Exponents 2 +
-

This exercise covers exponential arithmetic with rational bases and integer (primarily negative) exponents. This exercise covers all the material presented in the Level 2 Exponents video.

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@@ -60,24 +63,142 @@ VALS.sol_n VALS.sol_d + + reduce( BASEF_N, BASEF_D )[ 0 ] + reduce( BASEF_N, BASEF_D )[ 1 ]

\Large fracParens( BASE_N, BASE_D )^{EXP_SIGN+EXP}

SOL_N / SOL_D

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+
+

+ Any time we have a negative exponent, we can change it to a positive exponent if we flip the numerator and denominator: + [Why is that?] +

+
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+ The exponent tells us how many times to multiply by the base. If the exponent is + negative, then it tells us how many times to divide by the base. +

+

+ fracParens( BASE_N, BASE_D )^{\color{BLUE}{EXP_SIGN+EXP}} \quad = \quad \color{gray}{1} + \color{BLUE}{\div} fraction( BASE_N, BASE_D, false, true )\enspace +

+

+ Since dividing by a fraction is the same as multiplying by its reciprocal, we can replace the division with + multiplication: +

+

+ \hphantom{fracParens( BASE_N, BASE_D )^{EXP_SIGN+EXP}} \quad = \quad \color{gray}{1} + \color{GREEN}{\times} fraction( BASE_D, BASE_N, false, true )\enspace + \quad = \quad fraction( BASE_D, BASE_N, false, true, false, true )^{\color{GREEN}{EXP}} +

+

+ For more, check out the + negative exponent intuition video. +

+
+

+ fracParens( BASE_N, BASE_D )^{EXP_SIGN+EXP} \quad = \quad fracParens( BASEF_N, BASEF_D )^{EXP} +

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+
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= fracParens( BASEF_N, BASEF_D )^{EXP}

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+
+

+ To find fracParens( BASEF_N, BASEF_D )^{\color{BLUE}{EXP}}, + multiply cardinal( EXP ) round( BASEF_N/BASEF_D )s together. +

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+

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+

+ Any number to the first power is simply that number. +

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+
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-

= v

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+
+

+ Taking an exponent of a fraction is the same as taking the same exponent of the numerator and denominator: + [Why is that?] +

+
+

+ The exponent tells us how many times we're multiplying the fraction by itself. So in this case, we're + multiplying fraction( BASEF_N, BASEF_D, false, true, true ) by itself + cardinal( EXP ) times: +

+

+ fracParens( BASEF_N, BASEF_D )^{\color{BLUE}{EXP}} \quad = \quad + fraction( BASEF_N, BASEF_D, false, true ) + \times fraction( BASEF_N, BASEF_D, false, true )\enspace +

+

+ When we multiply fractions, we just multiply the numerators and denominators separately: +

+

+ \hphantom{fracParens( BASEF_N, BASEF_D )^{EXP}} \quad = \quad + \dfrac{jQuery.map( Array(EXP), function() { return reduce( BASEF_N, BASEF_D )[0] }).join("\\times ")} + {jQuery.map( Array(EXP), function() { return reduce( BASEF_N, BASEF_D )[1] }).join("\\times ")} + +

+

+ If we rewrite the numerator and denominator as exponents, we end up with the original fraction with the + numerator and denominator each raised to the original exponent: +

+

+ \hphantom{fracParens( BASEF_N, BASEF_D )^{EXP}} \quad = \quad + \dfrac{reduce( BASEF_N, BASEF_D )[0]^{\color{BLUE}{EXP}}} + {reduce( BASEF_N, BASEF_D )[1]^{\color{BLUE}{EXP}}} + +

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+

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+
+
+

Expand the exponents in the numerator and denominator into multiplication problems:

+

+ \hphantom{\qquad fracParens( BASEF_N, BASEF_D )^{EXP} \quad} = \quad + \dfrac{jQuery.map( Array( EXP ), function() { return BASEFS_N; }).join("\\times ")} + {jQuery.map( Array( EXP ), function() { return BASEFS_D; }).join("\\times ")} +

+
+
+

+ Multiply everything together: +

+

+ \hphantom{\qquad fracParens( BASEF_N, BASEF_D )^{EXP} \quad} = \quad + \dfrac{ + pow( BASEFS_N, I + 1 ) + jQuery.map( Array( EXP - I - 1 ), function() { return "\\times " + BASEFS_N; }).join("") + }{ + pow( BASEFS_D, I + 1 ) + jQuery.map( Array( EXP - I - 1 ), function() { return "\\times " + BASEFS_D; }).join("") + } +

+
+

+ Any number to the first power is simply that number. +

-

= frac( SOL_N, SOL_D )

+