MATHHX B
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1.2 Bogstavregning
Vi kommer til at regne meget med bogstaver, så det er vigtigt at få en god forståelse for bogstavregning fra starten af. Mange elever har svært ved bogstavregning. Det ved vi lærere godt, så spørg os, og vi hjælper!
-
Eksempel 1.2.1
Her er et udtryk med bogstaver:
\[\frac {a+7}{b}-a\]
Bogstaverne \(a\) og \(b\) står for tal, vi ikke kender. Derfor kan vi ikke regne udtrykkets værdi.
Lad os sige, at vi får at vide, at \(a=-2\) og \(b=5\). Nu kan vi regne værdien af udtrykket ved at erstatte \(a\) med \(-2\) og \(b\) med \(5\):
\(\seteqnumber{0}{1.}{0}\)
\begin{align*}
\frac {a+7}{b}-a & =\frac {-2+7}{5} -(-2)\\[5pt] & =\frac {5}{5}+2\\[5pt] & =1+2\\ & = 3
\end{align*}
Læg mærke til parentesen om \(-2\) i første linje.
Øvelse 1.2.1
Lad \(a=7\) og \(b=-3\). Regn:
-
a) \(a+2\)
-
b) \(a+b\)
-
c) \(a-b+1\)
1.2.1
-
a) \(9\)
-
b) \(4\)
-
c) \(11\)
Når vi regner med bogstaver, vil vi ofte undlade gangetegn. F.eks. vil vi skrive \(2a\) i stedet for \(2\cdot a\) og \(ab\) i stedet for \(a\cdot b\).
Øvelse 1.2.2
Lad \(a=-3\) og \(b=-1\). Regn:
Kender man ikke værdien af bogstaverne i et udtryk, kan man ikke regne udtrykkets værdi. Men nogle gange kan man forsimple det. Det kaldes at reducere.
-
Eksempel 1.2.4
Vi vil reducere udtrykket:
\[ a+ab+ba \]
Når man ganger, er rækkefølgen ligegyldig, så vi kan erstatte \(ba\) med \(ab\). Vi har nu:
\[ a+ab+ab \]
Vi ser, at vi nu har to \(ab\)’er, hvilket vi kan skrive som \(2ab\), så vi ender med:
\[ a+2ab \]
Øvelse 1.2.3
Reducer:
-
a) \(a+a+a\)
-
b) \(b+c-2a+a-c\)
-
c) \(b-x-x-b-5\)
Øvelse 1.2.4
Reducer:
-
a) \(ab-a+a-2b+b\)
-
b) \(bc+cb\)
-
c) \(ab+2+ab-3ba\)
I sidste afsnit lærte vi, at man skal regne indholdet af parenteser før alt andet. Men når der optræder bogstaver i regnestykket, kan man ikke altid regne parenteserne først. Så kan man i stedet benytte følgende regler til at hæve
(fjerne) parenteserne.
-
Eksempel 1.2.5
Her er nogle eksempler på brug af parentesregler:
\(\seteqnumber{0}{1.}{0}\)
\begin{align*}
&2(x+3)= 2x+2\cdot 3 = 2x+6 && \text {(gangeparentes)}\\ &a(b-c)= ab-ac && \text {(gangeparentes)}\\ &5+(a-2)=5+a-2= a+3 && \text {(plusparentes)}\\
&-(x+y)= -x-y && \text {(minusparentes)}\\ &-(2-a)=-2+a= a-2 && \text {(minusparentes)}
\end{align*}
Øvelse 1.2.5
Regn parenteserne:
-
a) \(3(x+y)\)
-
b) \((a-7)2\)
-
c) \(2+(x+y)-3\)
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d) \(-(v+w)\)
-
e) \(a(2b+b)-ba-(2a-2)\)
1.2.5
-
a) \(3x+3y\)
-
b) \(2a-14\)
-
c) \(x+y-1\)
-
d) \(-v-w\)
-
e) \(2ab-2a+2\)
Øvelse 1.2.6
To elever diskuterer, hvordan man regner udtrykket \(7(3+1)\).
Elev 1 siger: “Man skal starte med at regne det inde i parentesen og derefter gange med \(7\).”
Elev 2 siger: “Man skal gange parentesen ud og så reducere.”
Korrekt sprogbrug ()
I dette underafsnit vil du blive præsenteret for de fagord, du skal bruge for at udtrykke dig præcist. Det er markeret med (), hvilket
betyder, at det kan være udfordrende, og at man godt kan overleve grundforløbet uden at regne det.
Et udtryk er ”noget, som har en værdi”, som f.eks. \(2a+b\). (værdien af dette udtryk afhænger af værdien af \(a\) og \(b\)).
Et udsagn er ”noget, som har en sandhedsværdi”. Det kunne f.eks. være en ligning eller en ulighed.
-
Eksempel 1.2.7
Følgende er udsagn:
-
• \(2=5\)
-
• \(x+5=2\)
-
• \(2x+5<5(x+2)\)
Vi ser, at det første udsagn er falsk, mens sandhedsværdien af de to sidste udsagn afhænger af \(x\).
Øvelse 1.2.7
Bestem om følgende er udsagn eller udtryk:
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a) \(2=4+x\)
-
b) \(2\geq 7\)
-
c) \(x^2+4x+4\)
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d) \(5\)
1.2.7
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a) Udsagn
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b) Udsagn
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c) Udtryk
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d) Udtryk
En sum er et udtryk, som består af nogle størrelser lagt sammen. Her er et eksempel:
\[a+b+c\]
Størrelserne \(a\), \(b\) og \(c\) kaldes for led. En sum består altså af to eller flere led adskilt af plustegn.
En differens er det samme som et minusstykke:
\[a-b\]
Her kaldes \(a\) og \(b\) også for led.
Et produkt er et udtryk, som består af nogle størrelser ganget sammen. Det kunne være:
\[a\cdot b\cdot c\cdot d\]
Her kaldes \(a,b,c\) og \(d\) for faktorer. Et produkt består altså af to eller flere faktorer adskilt af gangetegn. Vi skriver ofte produkter uden gangetegn, så ovenstående produkt kunne
skrives som:
\[ abcd \]
En potens er et udtryk på formen:
\[a^{p}\]
Her kaldes \(a\) for grundtallet og \(p\) for eksponenten.
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Eksempel 1.2.8
Betragt udtrykket:
\[a(b+c)\]
Udtrykket er et produkt bestående af faktorerne \(a\) og \((b+c)\). Den anden faktor, altså \((b+c)\), er en sum bestående af de to led \(b\) og \(c\).
Øvelse 1.2.8
Forklar, hvordan følgende udtryk er opbygget:
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a) \(2ab\)
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b) \(a-ab\)
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c) \(3^{2b}\)
1.2.8
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a) Det er et produkt bestående af faktorerne \(2\), \(a\) og \(b\).
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b) Det er en differens bestående af leddene \(a\) og \(ab\). Det sidste led er et produkt bestående af faktorerne \(a\) og \(b\).
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c) Det er en potens, hvor grundtallet er \(3\), og eksponenten er \(2b\).