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Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
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Expressing the maximum of several variables using elementary functions

It's well-known that $$\max(a,b)=\frac{a+b+|a-b|}{2}.$$ Is there a (good) generalization to several variables? Of course $\max(a,b,c)=\max(a,\max(b,c))$ and…
Charles
  • 32,122
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5 answers

How to convert from a power of base two to a power of base 10?

I might have an extremely silly question: If I have a number, say $2^{32}$ and I need to convert to base 10, how should I do it? I know it should be $4 * 10^9$, but I do not know how did we get it. I understand that $10$ is $2^3 + 2$, but I cannot…
YohanRoth
  • 1,437
16
votes
5 answers

Intuition and derivation of the geometric mean

I've run through a bunch of searches, especially here on SO, but I simply couldn't find something that answers a question that has been on my mind lately. How was the geometric mean derived? What is the intuition behind it. Most simply use the final…
Rebuq
  • 163
15
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4 answers

Proof that there is no closed form solution to $2^x + 3^x = 10$

How can I prove that there is no closed form solution to this equation? $$2^x + 3^x = 10$$
Jaska
  • 1,291
15
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3 answers

Notation for rounding in equation

I'm wondering if there is a symbol or notation for Round to the nearest 10th For example, the area of a circle with a radius of 45 feet, rounded to the nearest square foot, could be written as, A = π45²sym Where sym is some symbol that means round…
Cnorwood7641
  • 151
15
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6 answers

If $2^x=0$, find $x$.

If $2^x=0$, find $x$. Solution: I know range of $2^x$ function is $(0,\infty)$. So $2^x=0$ is not possible for any real value of $x$ Hence, equation is wrong. We can't find value of $x$. Am I right? Please help me. Can $x$ be in…
rst
  • 2,031
15
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4 answers

Why do extraneous solutions exist?

I am currently in a Pre Calculus class at my High School. I have come across the concept of extraneous solutions, particularly when solving absolute value equations, radical equations, and logarithmic equations. My question is, why do these…
Nick
  • 315
15
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5 answers

How to average cyclic quantities?

Looking on Internet, I mostly found this definition: Given quantities on a cyclic domain D, first rescale the domain to [0;2$\pi$[, then, find $z_n$ the point on the unit circle corresponding to the $n$th value, and compute the average by: $$z_m =…
PierreBdR
  • 360
15
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2 answers

How to prove $(6666\ldots66)^2 + 8888\ldots88 = 4444\ldots44$ (with $n$ 6s and 8s, $2n$ 4s)

How to prove that, if $n$ is a positive integer, then $$ (\underbrace{666 \ldots 6}_{n \text{ copies of } 6})^2 + \underbrace{888 \ldots 8}_{n \text{ copies of } 8} = \underbrace{444 \ldots 4}_{2n \text{ copies of } 4}? $$
MathEnthusiast
  • 171
15
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7 answers

Algebraic Identity $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$

Prove the following: $\displaystyle a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$. So one could use induction on $n$? Could one also use trichotomy or some type of combinatorial argument?
NebulousReveal
  • 13,935
14
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8 answers

Whats the rule for putting up a plus-minus sign when taking under root?

Given a simple equation.... $\ (x+1)^2 =21 $ if we take the under root of both sides , we get $\ x+1 = \pm \sqrt{21} $ why dont we get a $ \pm $ on the left hand side ?
explorest
  • 415
14
votes
3 answers

Why multiply first?

Why do we multiply/divide first, and then add/subtract later? I mean, what I'm curious about is that is this a universal rule, or a man-decided rule? Also how would you decide on which to operate first? For example, if we were to be visited by…
akinuri
  • 1,353
14
votes
3 answers

Finding the real solutions to $16^{x^{2} + y } + 16^{y^{2}+ x} = 1$

We have , $16^{x^{2} + y } + 16^{y^{2}+ x} = 1$ , then we have to find all the real values of $x$ and $y$.I tried this question but i am not able to proceed because I am not able to simplify this expression to an extent that it could be solved.
Achal Kumar
  • 503
14
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11 answers

Solve $4^{9x-4} = 3^{9x-4}$

I am having some trouble trying to solve $$4^{9x-4} = 3^{9x-4}$$ I tried to make each the same base but then I'm becoming confused as to what to do next. These are the steps I took: $$\begin{align} 4^{9x-4} &= 3^{9x-4} \\ \log_4(4^{9x-4}) &=…
Jeel Shah
  • 9,306
14
votes
4 answers

Number of surjective functions from A to B

Am I on the right track? I am not sure about my reasoning... Number of surjective functions from $A$ to $B$ $$A = \{1,2,3,4\} ; B = \{a,b,c\}$$ We must count the surjective functions, meaning the functions for which for all $b \in B$, $\exists~a \in…
Justin D.
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