Mathematics Stack Exchange
Mathematics Stack Exchange

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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Rationalizing radicals

Rationalize (i.e., get rid of radicals in the denominator) in $\dfrac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}}$, $\dfrac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}$, and, in general $\dfrac{1}{\sum_{i=1}^n \sqrt{a_i}}$. I have submitted my work so far as an…
marty cohen
  • 107,799
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Comparing $\frac {9}{\sqrt{11} - \sqrt{2}}$ and $\frac {6}{3 - \sqrt{3}}$ (without calculator)

We want to compare the following two numbers: $$x = \frac {9}{\sqrt{11} - \sqrt{2}} \quad\text{and}\quad y = \frac {6}{3 - \sqrt{3}}$$ My attempts so far: I multiply both numerator and denominator of $x$ by $\sqrt{11} + \sqrt{2}$ so I get: $$x =…
Pradeep Suny
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Explaining an algebra step in $ \frac{n^2(n+1)^2}{4} + (n+1)^3 = \frac{(n+1)^2}{4}(n^2+4n+4)$

I have encountered this step in my textbook and I do not understand it, could someone please list the intermediate steps? $$ \frac{n^2(n+1)^2}{4} + (n+1)^3 = \frac{(n+1)^2}{4}(n^2+4n+4). $$ Thanks,
Jason
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If $m+n=5$ and $mn=3$, find $\sqrt{\frac{n+1}{m+1}} + \sqrt{\frac{m+1}{n+1}}$?

It is known that $m+n=5$ and $mn=3$. So what is the value of: $$ \sqrt{\dfrac{n+1}{m+1}} + \sqrt{\dfrac{m+1}{n+1}} $$ I think we're suppose to solve for the system of equations first, but I'm not getting any results that's useful.
Ron Davidson
  • 253
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For $x, y, z \in \mathbb{R}$, $(x^y)^z = x^{yz}$?

Is it always true that for $x, y, z \in \mathbb{R}$, $(x^y)^z = x^{yz}$, whenever both expressions are defined? Assume all are nonzero. I think this isn't true in general, because for instance $$-1 = (-1)^{2/2}$$ but $$((-1)^2)^{1/2} = 1^{1/2} =…
user81412
  • 71
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Algebra simplification in mathematical induction .

I was proving some mathematical induction problems and came through an algebra expression that shows as follows: $$\frac{k(k+1)(2k+1)}{6} + (k + 1)^2$$ The final answer is supposed to be: $$\frac{(k+1)(k+2)(2k+3)}{6}$$ I walked through every…
Evy Lino
  • 85
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Did I get the correct answer, and if so why?

The question asks to find the equation of the line passing through $(0,2)$ and just touching $y=(x+2)^{2}$ where $x>0$ I let the equation of the line have gradient $m$ therefore the line is $y=mx+2$ and for the lines to meet: $$mx+2=(x+2)^{2}$$…
Johnmgee
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how to solve this multivariate quadratic equation?

Any hope to acquire an analytic solution to such equations: Solve: $$\sum_{j=1}^n a_{ij} x_i x_j = b_i$$ for $i=1,\ldots,n$, where $a_{ij}$'s and $b_i$'s are known constants and $x_i$'s are unknowns to be solved. Thanks a lot! P.S. Thanks for…
yuanz07
  • 145
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solving $1+\frac{1}{x} \gt 0$

In solving a larger problem, I ran into the following inequality which I must solve: $$ 1+\frac{1}{x} \gt 0.$$ Looking at it for a while, I found that $x\gt 0$ and $x\lt -1$ are solutions. Please how do I formally show that these are indeed the…
Gorg
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$\frac{(x + \sqrt{x}) - (x-\sqrt{x})}{\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}} = \frac{2}{\sqrt{1+\frac{1}{\sqrt{x}}}+\sqrt{1-\frac{1}{\sqrt{x}}}}$?

According to an example in my text book: $$\frac{(x + \sqrt{x}) - (x-\sqrt{x})}{\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}} = \frac{2}{\sqrt{1+\frac{1}{\sqrt{x}}}+\sqrt{1-\frac{1}{\sqrt{x}}}}$$ I don't see how this works. The closest I can get…
Quispiam
  • 851
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Substitution problem

My question is something I've been thinking about for some time now. Q: Why is it possible to make substitutions or change in variables ? I mean, how do I know which substitutions are allowed ? For example when we use Vieta's formulas to vanish with…
Bandeira
  • 73
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Finding the remainder when $5^{55}+3^{55}$ is divided by $16$

Find the remainder when $5^{55}+3^{55}$ is divided by $16$. What I try $a^{n}+b^{n}$ is divided by $a+b$ when $n\in $ set of odd natural number. So $5^{55}+3^{55}$ is divided by $5+3=8$ But did not know how to solve original problem Help me please
jacky
  • 5,194
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Solving $\frac{\ln(x)\ln(y)}{\ln(1-x)\ln(1-y)}=1$ for $y$

I'm trying to solve for $y$ in terms of $x$ for the expression below. $$\frac{\ln(x)\ln(y)}{\ln(1-x)\ln(1-y)}=1$$ First I multiplied both sides by $$ \frac{\ln(1-x)}{\ln(x)} $$ to get $$ \frac{\ln(y)}{\ln(1-y)}=\frac{\ln(1-x)}{\ln(x)} $$ but I…
John Zimmerman
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Separation of geometric sequence do not have equal sum.

Suppose we remove some terms (taking at least 1 item and leaving at least 2 items) from the geometric sequence, $$1, k, k^2, \cdots, k^{n}$$ (with $k>2$) and separate the remaining terms into two groups. Prove these two group's sum can never…
Baker5680
  • 451
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Why is $\sum\limits_{i=1}^n i = \frac{n(n+1)}{2}$?

Possible Duplicate: Proof for formula for sum of sequence $1+2+3+\ldots+n$? I was just messing around in Haskell: let fnc x = (sum [1..x], x * (x - 1) / 2) fnc 12 >> (78.0, 66.0) (I was just messing around after reading The Mythical Man Month:…
user9912
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