January 9th, 2015, 02:03 PM  #1 
Newbie Joined: Oct 2014 From: england Posts: 23 Thanks: 0  Confused with indices
Hi, so recently I've been learning about powers and indices. Were currently learning how to work out algebraic powers. I was given this question... 3y^2+3 = 9^2y I'm confused as to how the +3 has become a power in the first place? e.g would it have been 3y^2 * 27 = 9^2y and then 3y^2 * 3^3 = 9^2y ? Thanks for any advice. Last edited by skipjack; January 9th, 2015 at 03:45 PM. 
January 9th, 2015, 02:37 PM  #2  
Math Team Joined: Jul 2011 From: Texas Posts: 2,982 Thanks: 1575  Quote:
Is it possible you meant $\displaystyle 3y^2+3 = 9^{2y}$ ? Maybe something else? Recall your order of operations and use some grouping symbols ... parentheses would be nice ... to clarify your equation. Thanks. Last edited by skipjack; January 9th, 2015 at 03:45 PM.  
January 9th, 2015, 03:29 PM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Hello, nicevans1! Some parentheses would be welcome. I'll guess at what you meant. Quote:
We have: $\:3^{y^2+3} \:=\:9^{2y} \quad\Rightarrow\quad 3^{y^2+3} \:=\:(3^2)^{2y} \quad\Rightarrow\quad 3^{y^2+3} \:=\:3^{4y}$ Hence: $\:y^2+3 \:=\:4y \quad\Rightarrow\quad y^2  4y + 3 \:=\:0 \quad\Rightarrow\quad (y1)(y3) \:=\:0$ Therefore: $\:y \:=\: 1,\,3$  
January 13th, 2015, 11:04 AM  #4  
Newbie Joined: Oct 2014 From: england Posts: 23 Thanks: 0  Quote:
Quote:
So first can you explain (as this question really took me up the wrong alley) What is actually an indice and what is not? Is the first Y a power and if not why is it the same size in text as the y at end? Also why is that 2 smaller than the 3 in terms of size of text? The 2 is also raised higher than the rest of the powers, why?? I honestly presumed it was a mis print in the book and its really thrown me. Thanks for your time Last edited by greg1313; January 13th, 2015 at 11:17 AM.  
January 13th, 2015, 02:14 PM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
In the United States, an "index" is simply a subscript or superscript labeling a variable or constant. But our English friends use it to denote what we would call an "exponent". In the very simplest case, when it is a positive integer, it just tells how many times the base is multiplied by itself: , , , etc. From that you can, perhaps by induction on n and m, that and . Those are nice properties so we extend to other numbers. If, for example, m= 0, we have . In order to have we must define . Now, to extend to negative integers, look at . We must define for n> 0. Next, look at m= 1/n. . In order for the rule for exponents of the form , then we must define to be . To continue to all real numbers, we have to switch from 'algebraic' to 'analytic' properties. A function is said to be "continuous at x= a" if and only if is a sequence of numbers that converge to a, then the sequence converges to f(a). (That is one of many equivalent definitions of "continuous".) If x is any real number, then there exist a sequence of rational numbers, that converges to x so, in order to have be continuous, we define where is any sequence of rational numbers that converges to x. 
January 13th, 2015, 03:27 PM  #6 
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191 
Country Boy, Don't try to use .  They don't work. Instead use [ MATH ], [ /MATH ], but take out the spaces between "[" and "M" and "H" and "]." Also, take out the spaces between "[" and "/" and "H" and "]" again. 
January 13th, 2015, 04:42 PM  #7  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
You don't need any of Country Boy's definitions if you start from the other end. We want(ed) a function such that $f(xy) = f(x) + f(y)$ to make multiplication and division easier. It turns out that the only nontrivial functions that do this are of the form $$L(x) = c\int_1^{x} \tfrac1t \, \mathrm d t \qquad \text{where $x \ne 0$}$$ So, if we define the natural logarithm to be $$\log x = \int_1^x \tfrac1t \, \mathrm d t \qquad \text{where $x \gt 0$}$$and define the number $\mathrm e$ to be the (unique) solution to $\log x = 1$, we can then define $\mathrm e^x$ to be the inverse function of $\log x$. That is$$y = \log x \implies \mathrm e^y = x$$ This defines $\mathrm e^x$ for all (real) $x$ without any messing around. It also gets us $a^x = \mathrm e^{x \log a}$ for any positive $a$, and all the nice properties that we with exponentiation to have can be proved. We get logarithms in other bases either as inverse functions to $a^x$ so that $$a^x = y \implies \log_a y = x$$or using the logarithmic identity$$\log_a x = {\log x \over \log a }$$ I quote Country Boy's post in its entirety with tags corrected. Quote:
Last edited by v8archie; January 13th, 2015 at 05:07 PM.  
January 14th, 2015, 01:59 PM  #8  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  Quote:
 
January 18th, 2015, 01:22 PM  #9  
Newbie Joined: Oct 2014 From: england Posts: 23 Thanks: 0  Quote:
When I said what is an indice, I meant in terms of the question I posted about. So what Im asking is. What is an exponent in the question as all the numbers and letters seem a different levels? So im confused as to what is an exponent and what is not! 3y^2+3=9^2y Thankyou  
January 19th, 2015, 08:33 AM  #10 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित 
you need to use latex to write the expression in appropriate form like you write in a paper. If you don't know latex you can use brackets ( ) for understanding. like $\displaystyle 3^{y^2+3}=9^{2y} $ can be written as 3^(y^2+3)=9^(2y) The exponent of a number says how many times to use that number in a multiplication, its written at the top of the number or after the symbol ^ if only one constant or variable is used as exponent, brackets ( ) are not required. If more than one constant or variable is used as exponent we need brackets ( ) for clear understanding. 

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