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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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Is there a number $a$ such that $a^2$ is irrational, but $a^4$ is rational?

This problem is from Spivak Calculus, now you immediately think $a^2 = \sqrt{2}$ and $a^4 = 2$, so the answer to the problem is yes. But, the way Spivak has developed his book: So far we only have the algebraic axioms (associativity, commutativity,…
Édgar Sánchez Gordón
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Simpler or shorter way to simplify expression $(16^{2} \times 64^{3})\div1024^{2}$ for a power

I am studying about powers for a discipline in college and the teacher asked me to simplify the following expression to transform it into the form of a single power, $$ (16^{2} \times 64^{3})\div1024^{2} $$ I can simplify to, $$ 2^{6} $$ But, take…
gato
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Simplifying expression: $1 - \frac{1}{ (1 + a) / (1 - a)}$

Correct Answer is $a$ My attempt: $$1 - \frac{1}{ (1 + a) / (1 - a)}= 1 - \frac{1 - a}{1 + a} $$ multiply $(1 + a)$ on num and dem for the first term. $$=\frac{1 + a}{1 + a} - \frac{1 - a}{1 + a}$$ combine and subtract. $$=\frac{2a}{1 + a}$$ How…
user9995331
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Number of copper atoms in $1\mbox{cm}^3$ of copper

I'm starting my physics class and I'm really rusty on my conversions and stoichiometry, The mass of a copper atom is $1.37\cdot 10^{-25}$ kg, and the density of copper is $8920 \mbox{kg/m}^3$. I would ask this in physics.se, but the problem is…
OghmaOsiris
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How to solve equations involving multiple floors of the same variable?

I want to solve equations of this type: $$\lfloor1x\rfloor+\lfloor2x\rfloor+\lfloor3x\rfloor+\lfloor4x\rfloor+\lfloor5x\rfloor=10$$
Vivek Sivaramakrishnan
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How to solve $\sqrt{49-x^2}-\sqrt{25-x^2}=3$?

I recognize the two difference of squares: $49-x^2$ and $25-x^2$. I squared the equation to get: ${49-x^2}-2(\sqrt{(49-x^2)(25-x^2)})+{25-x^2}=9$ However, I can't quite figure out how to remove the root in the middle. Any help is appreciated.
Kohei Sanno
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Summation of series (general term)

Hi, I need help with this. I haven't got a clue how to solve this question. I'm familiar with AP, GP, HP, AGP, some special series... I tried, but couldn't figure out how to approach this question. I have attached two pictures, one is the…
4d_
  • 570
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What happens to roots of a function when deriving?

While studying, I stumbled upon a statement that all the roots of $q(x) = x^{(p^n)} - x$, where p is a prime number are different, because otherwise $q$ would share some factors with its derivative. This lead me to wonder, what happens to the roots…
klemen kobau
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Find an equation of the plane

I have come across a difficult problem in my textbook. The problem ask: Find an equation of the plane that passes through the line of intersection of the planes $x-z=1$ and $y+2z=3$ and is perpendicular to the plane $x+y-2z=1$. So far I found the…
jascal
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Rationalizing denominator - why?

In many Algebra textbooks, why rationalizing the denominator is defined to be the simplest form ? I try to understand it but I cannot.Why is $\frac{\sqrt 3}{3}$ simpler than $\frac1{\sqrt 3}$?
Ahmed M. Elsonbaty
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find the minimum value given constraints

Let $z$ be a complex number such that $\dfrac{z-i}{z-1}$ is purely imaginary. Find the minimum value of $|z-(2+2i)|$. Source: ISI 2017 BMATH UGA $$$$ Attempt: Since $\dfrac{z-i}{z-1}$ is purely…
User1234
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Arithmetic Sequence Problem I Can't Figure it Out

$$\frac{1}{\sqrt a_1 + \sqrt a_2} + \frac{1}{\sqrt a_2 + \sqrt a_3} + \cdots + \frac{1}{\sqrt a_{n-1} + \sqrt a_n} = \frac{n-1}{\sqrt a_1 + \sqrt a_n}$$ I need to prove this; I don't know how to get from left side terms to the right side term? What…
ronenp88
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Range of $k$ for which equation has positive roots

Range of $k$ for which both the roots of the equation $(k-2)x^2+(2k-8)x+3k-17=0$ are positive. Try: if $\alpha,\beta>0$ be the roots of the equation. Then $$\alpha+\beta=\frac{8-2k}{k-2}>0\Rightarrow k\in(2,4)$$ And…
DXT
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Algebra question involving fractions

How would I perform the indicated operation. $$\frac{t+2}{t^2+5t+6}+\frac{t-1}{t^2+7t+12}-\frac{2}{t+4}.$$ I simplified it to $$ \frac{t+2}{(t+3)(t+2)} + \frac{t-1}{(t+4)(t+3)}-\frac{2}{t+4}. $$ Then I did the lowest common denominator but I still…
Fernando Martinez
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The square of the solution of the equation...

The square of the solution of the equation $x\sqrt{7} + \sqrt{8 - 3\sqrt{7}} - \sqrt{8 + 3\sqrt{7}} = 0$ is equal to: ... $x\sqrt{7} + \sqrt{8 - 3\sqrt{7}} - \sqrt{8 + 3\sqrt{7}} = 0$ $\implies x\sqrt{7} = \sqrt{8 + 3\sqrt{7}} - \sqrt{8 -…
Brian Blumberg
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