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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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Limits of Exponent Laws

I have recently learned (discovered) that the exponent law $b^{mn} = {(b^m)}^n$ is not universally applicable. To demonstrate, if it were we could conclude that $(-1)^{\frac{3}{2}}$ (or by extension -1 to any power) is equal to…
Anomaly
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Factoring $x^4 + 4x^2 + 16$

I was putting together some factoring exercises for my students, and came across one that I am unsure of how to factor. I factored $x^6 - 64$ as a difference of squares, and then tried it as a difference of cubes, but was left with $(x^2 - 4)(x^4 +…
drawnonward
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no. of distinct real roots of the equation $x^2=x\sin x+\cos x$

$(1)$ The no. of Distinct real solution of the equation $x^4-4x^3+12x^2+x-1=0$ $(2)$ The no. of distinct real roots of the equation $x^2=x\sin x+\cos x$. $\bf{My\; Try}$ For $(1)$ one:: Let $f(x) = x^4-4x^3+12x^2-x+1$. Then $f'(x) =…
juantheron
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Evaluation of $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{1}{3^{i+j+k+l}}\;\;,$ Where $i\neq j \neq k\neq l$

Evaluation of following Infinite Geometric series. $(a)\;\; \displaystyle \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{1}{3^{i+j}}\;\;,$ Where $i\neq j\;\;\;\;\;\;\;\;\;\; (b)\;\; \displaystyle…
juantheron
  • 53,015
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prove that $a^3 - b(b+1) = (a-2)(c^2 + 1) + 2$ has infinitely many solutions

How do I prove that $$a^3 - b(b+1) = (a-2)(c^2 + 1) + 2$$ has infinitely many solutions if a, b and c are natural numbers? I have opening the brackets and moving all the terms to one side which gets rid of the constant 2. I have also tried…
Life is Hard
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Calculating the percent of a person's life that a certain time takes up?

A while ago I read an article on Cracked.com about why you wouldn't want to be immortal. Among many other reasons was that time would speed up for you after so many years and it would make every second seem fleeting. Relative to a year old, each day…
mowwwalker
  • 1,201
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If $a+\frac1a=\sqrt3$ then $a^4+\frac1{a^4}=\ ?$

If $a+\frac1a=\sqrt3$ then $a^4+\frac1{a^4}=\ ?$ Can someone please explain to me how to solve this? because I tried everything I know and it didn't work. P.S: I'm in 8th grade so no quadratic formula.
Yassir
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Product of numbers that remains invariant repeatedly when adding one to all of them

Eugene wrote a 100 numbers on a board. He added 1 to each number and the product didn't change. He did the same thing k times, each time the product didn't change. What is the maximum k? I guess the answer is 99 and that happens if the numbers are…
MelanieJohansson
  • 485
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Basic Math, exponents and algebra

I have the equation $$\frac{x_1^{-\frac{1}{2}}}{{x_2^{-\frac{1}{2}}}} = p_l/p_2$$ How do I get $x_2$ on its own? I have $$x_2^{-\frac{1}{2}} = \frac{p_2(x_1^{-\frac{1}{2}})}{p_1}$$ And if you have a useful link that reviews this info, it would be…
Sara
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systems of linear equations intuition

I want to know why in a system of linear equations I'm allowed to sum or subtract the equations. I can't get the intuition of why I can do that to solve for the equations.
kprincipe
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How fast does the water level decrease in a cylindrical tank?

Is this solution correct? What I know is that the volume of the tank is $V = \pi r^2 h$, where r and h are in meter. Water is drained by a rate of $2,7\frac{m^3}{min}$. How fast does the water level decrease in this tank? So if I set $H(t)$ to be…
Frank Vel
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I need help solving $3e^{2x}-1=\frac{1}{2}$

I am trying to solve $3e^{2x}-1=\frac{1}{2}$. Here is my work: $3e^{2x}-1=\frac{1}{2}$ $3e^{2x} =\frac{1}{2}+1$ $e^{2x} =\frac{1.5}{3}$ $\ln{e^{2x}} =\ln{(\frac{1}{6})}$ $2x =\ln(\frac{1}{6})$ $x =\frac{\ln(\frac{1}{6})}{2}$ $x …
Hein Terblanche seemykungfu
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maximize product of two numbers formed with the digits $1$ to $9$

You are given the digits from $1$ to $9$. You can form two numbers by concatenating them, for example, $975123$ and $864$, and then take the product of the two resulting numbers. Find how to maximize the product. To give you a hint, the answer is…
Qiang Li
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Make x the subject given the formula for y

I am given the following formula: $$y=\frac{x}{a}+\sqrt{\frac{x}{b}}.$$ I want to make x the subject. I rearranged the equation and got to: $$y^{4}=x(\frac{y^{2}}{2}+2y^{2})-x^{2}.$$ and I don't know where to go from here. May be this is the wrong…
user4010936
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Is this parametric curve a rotated parabola?

I have a parametric curve that is quadratic in both x(a) and y(a). Specifically, $$x(a) = -a^2 +150a$$ $$y(a) = -3a^2 + 500a$$ It looks like a rotated parabola. How can I be sure? See my Maple worksheet. Thanks, Matt
user145600
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