Question

What is 2+2?



Exponents Tutorial
(Sorry for the trick heading!)

Answer 1

\(\hspace{10pt}\Huge\sf \color{#001d5a}{Exponents~Tutorial}\)

\(\hspace{5pt}\large\sf Hello~and~welcome~to~StudyGurl14's~Exponents~Tutorial!\)

\(\sf\color{blue}{First~off}\), exponents are a way to express repeated multiplication without the hassle of actually writing out everything. \(\sf\color{purple}{For~example}\), \(2\times2\times2~or~{(2)(2)(2)}\) can be written with an exponent like this: \(\large2^3\)
The \(\large 2\) in this expression is called the \(\sf\color{green}{"base"}\), a.k.a. the number being multiplied. The superscript \(3\) above the \(\large 2\) is called the \(\sf\color{green}{"exponent"}\) or \(\sf\color{green}{"power"}\), and represents the amount of times the 2 is being multiplied. You can read the above term as \(\sf\color{green}{"two~raised\ to\ the\ third\ power"}\), or as \(\sf\color{green}{"two\ to\ the\ third"}\). (Of course, the most popular exponents, 2 and 3, are known as \(\sf\color{green}{"squared"}\) and \(\sf\color{green}{"cubed"}\), respectively. Which means the above example could also be read as \(\sf\color{green}{"two~cubed"}\)).
Exponents are especially helpful in the cases when the base value is not a number, but instead is a variable. \(\sf\color{purple}{For~example}\), \(xxxx\) can be expressed like this: \(\large x^4\).

\(\sf\color{blue}{Note: }\) Exponents with negative numbers as bases are a little more complicated than bases that are positive numbers. But don't worry. You can learn about that easily. See the below \(\sf\color{purple}{examples:}\)
\(\large\ -3^2=-(3)(3)=-9\)
\(\large\ (-3)^2=(-3)(-3)=9\)
Do you see the difference? Always assume that the exponent only applies only to the number to the direct left of it, unless parentheses indicate otherwise.
\(\sf\color{purple}{Another\ example\ to\ demonstrate\ this:}\)
\(\large 3x^2=(3)(x)(x)\)
\(\large (3x)^2=(3x)(3x)=9x^2\)

\(\sf\color{blue}{There\ are\ some\ mathematical\ rules\ that\ deal\ with\ exponents.}\) These can be very helpful when you are dealing with a seemingly very complicated mathematical expression. Using the rules that I will discuss below, you can turn a complicated expression into something much simpler and manageable.

\(\sf\color{blue}{Exponent~Rule~\#1: }\) \(\large x^1=x\)
\(\sf\color{purple}{Example: }\) \(\large 3^1=3\)
The same applies to variables.
\(\sf\color{purple}{Example: }\) \(\large y^1=y\)
\(\large Why?\) The exponent represents the amount of time the base is “shown” or “present” in the expression. Because the exponent is 1, the base is only shown once, and can just be written as itself, without the exponent.

\(\sf\color{blue}{Exponent~Rule~\#2: }\) \(\large 1^n=1\)
\(\sf\color{purple}{Example: }\) \(\large 1^{95,486,245}=1\)
\(\large Why?\) One times itself is ALWAYS one, no matter how many times you multiply it by itself. Meaning \(\large 1^2 = 1\), and 1 to the billionth power also equals 1.

\(\sf\color{blue}{Exponent~Rule~\#3: }\) \(\large x^0=1, where~x\neq 0\)
\(\sf\color{purple}{Example: }\) \(\large 12,345^0=1\)
The same applies to variables.
\(\sf\color{purple}{Example: }\) \(\large z^0=1\) (assuming \(z\neq 0\))
\(\large Why?\) I honestly have \(\sf\color{red}{no\ idea}\). Sorry! It is just a known rule that I follow just 'cause it's a rule.

\(\sf\color{blue}{Exponent~Rule~\#4: }\) \(\large (x^m)(x^n)=x^{m+n}\)
\(\sf\color{purple}{Example: }\) \(\large (2^2)(2^3)=2^{2+3}=2^5=32\)
\(\large Why?\) Remember how exponents are just another way of expressing repeated multiplication? Well, \(\large (2^2)(2^3)\) is just another way to say \(\large (2\times2)(2\times2\times2)\), which can also be written as \(\large (2)(2)(2)(2)(2)\). Notice how 2 is being multiplied by itself 5 times? It is because of this that you can simplify it into \(\large 2^5\)
You can also do the same with exponents that have bases that are variables, as long as the bases are the same variable.
\(\sf\color{purple}{Example: }\) \(\large (y)(y^5)=y^{1+5}=y^6\)
\(\sf\color{red}{Warnings: }\)
\(\large 4^2\times 2^3 \neq 8^5\)
\(\large 4^2 +4^3\neq 4^5\)

\(\sf\color{blue}{Exponent~Rule~\#5: }\) \(\large x^m \div x^n=x^{m-n}\)
\(\sf\color{purple}{Example: }\) \(\large 4^5 \div 4^3=4^{5-3}=4^2=16\)
\(\large Why?\) Division if the opposite of multiplication, like a negative number is the opposite of a positive number. I can't really explain it as well as I can show it:
\(\large 4^5 \div 4^3=\frac{4^5}{4^3}=\frac{(4)(4)(4)(4)(4)}{(4)(4)(4)}=\frac{\cancel{(4)}\cancel{(4)}\cancel{(4)}(4)(4)} {\cancel{(4)} \cancel{(4)} \cancel{(4)}}=(4)(4)=4^2\)
The same applies to variables.
\(\sf\color{purple}{Example: }\) \(\large c^{10} \div c^4=c^{10-4}=c^6\)

\(\sf\color{blue}{Exponent~Rule~\#6: }\) \(\large (x^m)^n=x^{m(n)}\)
\(\sf\color{purple}{Example: }\) \(\large (4^2)^2=4^{2(2)}=4^4=256\)
\(\large Why?\) \(\large (4^2)^2\) is another way to express \(\large (4^2)(4^2)\), which in turn is another way to express \(\large (4)(4)(4)(4)\). There are 4 4s, so it is equal to \(\large 4^4\)
The same applies to variables.
\(\sf\color{purple}{Example: }\) \(\large (a^2)^4=a^{2(4)}=a^8\)

\(\sf\color{blue}{Exponent~Rule~\#7\ (extension\ of\ \#6): }\) \(\large (x^my^n)^p=x^{m(p)}y^{n(p)}\)
\(\sf\color{purple}{Example: }\) \(\large (4^25^3)^2=4^45^6=235(15625)=3,671,875\)
\(\large Why?\) \(\large (4^25^3)^2\) is another way to express \(\large (4^25^3)(4^25^3)\), which in turn is another way to express \(\large \color{orange}{(4)(4)}(5)(5)(5)\color{orange}{(4)(4)}(5)(5)(5)\). Notice how there are 4 4s (colored in orange) and 6 5s (in black) multiplied together, so it is equal to \(\large 4^45^6\)
The same applies to variables.
\(\sf\color{purple}{Example: }\) \(\large (xy)^5=x^5y^5\)(remember the invisible exponent 1s)
\(\sf\color{red}{Warning: }\)
\(\large (2^2+3^2)^2\neq 2^4+3^4\)

\(\sf\color{blue}{Exponent~Rule~\#8\ (further\ extension\ of\ \#6): }\) \(\Large (\frac{x^m}{y^n})^p=\frac{x^{m(p)}}{y^{n(p)}}, where~y\neq0\)
\(\sf\color{purple}{Example: }\) \(\Large (\frac{3^2}{2^3})^2=\frac{3^{2(2)}}{2^{3(2)}}=\frac{3^4}{2^6}=\frac{81}{64}\approx 1.27\)
\(\large Why?\) \(\Large (\frac{3^2}{2^3})^2\) is another way to express \(\Large (\frac{3^2}{2^3})(\frac{3^2}{2^3})\), which in turn is another way to express \(\Large \frac{(3)(3)(3)(3)}{(2)(2)(2)(2)(2)(2)}\). Notice how there are 4 3s in the numerator and 6 2s in the denominator multiplied together, so it is equal to \(\large\frac{3^4}{2^6}\)
The same applies to variables.
\(\sf\color{purple}{Example: }\) \(\Large (\frac{a^2}{b})^3=\frac{a^6}{b^3}\)(assuming \(b\neq0\))

\(\sf\color{blue}{Exponent~Rule~\#9: }\) \(\large x^{-m}=\frac{1}{x^m}\)
\(\sf\color{purple}{Example: }\) \(\large 2^{-3}=\frac{1}{2^3}=\frac{1}{8}\)
\(\large Why?\)
The same applies to variables.
\(\sf\color{purple}{Example: }\) \(\large b^{-1}=\frac{1}{b}\)(assuming \(b\neq0\))


\(\large\sf\color{purple}{Examples\ of\ Problems: }\)
Will be posting later when I have more time.


\(\Large\sf\color{red}{Final~Notes: }\)
\(\large\sf Many\ thanks\ to\) @thomaster \(\large\sf and\ his~\href{ http://openstudy.com/study#/updates/52e12f56e4b0942cc9de719e }{\color{blue}{Lengendary\ LaTeX\ Tutorial}}\)
\(\large\sf for\ helping\ me\ with\ the\ LaTeX\ aspects\ of\ this\ tutorial.\)

\(\large\sf Please\ don't\ post\ any\ questions\ regarding\ this\ tutorial\ in\ this\ thread.\)
\(\large\sf Instead,\ please\ open\ your\ question\ in \ your\ own\ thread\ and\ tag\ me\ to\ it.\ Thx.\)

\(\large\sf If\ you\ see\ any\ question\ marks,\ refresh\ the\ page.\)

\(\large\sf\color{red}{If\ I\ Made\ ANY\ Mistakes\ PLEASE,\ please,\ please\ message\ me. Thanks.}\)

Answer 2

Ah, the link didn't work and the last sentence was cut off, but whatever. :)

Answer 3

this look amazing!

Answer 4

Thanks. :)

Answer 5

submit it on tutorial blog when you'll recieve 5+ medals. they might not review you if you have <5 medals.

Answer 6

Okay, thanks for the heads up about the medal part.

Answer 7

2 + 2 = 4

Answer 8

lol, I know @HotPinkGirl

Answer 9

then why did you ask?

Answer 10

I didn't. Not really anyway.

Answer 11

Look up^ and read the title again

Answer 12

Nicely done! :D

Answer 13

Good job! :D

Also they do not look at how many medals you got... I know of like 2 tutorials that only had 1 medal, until I gave it the 2nd medal. And I doubt those tutorials have 5 medals yet. So don't worry about how many medals you have gotten, just submit it as soon as you can, and hopefully you will make it in the top 10! :D

Answer 14

A good tutorial. You added all the formulas that I know. Keep it up! @StudyGurl14

Answer 15

Thanks @EclipsedStar @TheSmartOne @confluxepic

Answer 16

No problem. :)

Answer 17

2+2=5

Answer 18

ohh that's really gO_Od
perfect!!!
gO_Od job!!! @StudyGurl14
i was like huh @StudyGurl14 post about 2+2 ? :O

Answer 19

lol, thx @Nnesha

Answer 20

its 4!!
:)

Answer 21

:)

Answer 22

Liked it!
Very useful :)