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
4
votes
4 answers

Determine whether $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational

I need some help with this problem: the task is to determine if the number $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational or not. Unfortunately I barely have an idea how to start and hence would appreciate not a solution but rather a way to get…
mooculus
  • 41
4
votes
1 answer

Find an equation for u and v

We have the function $x^3 + px + q = 0$, where $p$ and $q$ are known real numbers and $x$ is an unknown real number. Put $x = u + v$ and write it out. If $3uv+p=0$, can you find another equation for $u$ and $v$? So, for the first step: $u^3 +…
Preet Singh
  • 73
4
votes
3 answers

Solving $(1.44)^t=t^{1.44}$

I've been trying to solve the equation $(1.44)^t=t^{1.44}$, but other than the obvious solution ($t=1.44$) I haven't had much luck manipulating this into something useful. By taking the log of both sides I'm able to get $\dfrac{\ln t}{t}=\dfrac{\ln…
steve
  • 303
4
votes
3 answers

How does one simplify the expression $\sqrt[3]{2 \sqrt{2}}\left(\sqrt{2-\sqrt{3}}+\sqrt{2+\sqrt{3}}\right)$

I don't know how to solve this problem. What I know is $\sqrt[3]{2 \sqrt{2}}=\sqrt{2}$. But I don't know how to continue.
Infinity
  • 161
4
votes
4 answers

Prove $f(x) = 3^x$ is not onto.

Give a counterexample to prove $f(x) = 3^x$ is not onto. A function is onto if for all $y$ in the codomain, there exists an $x$ in the domain such that $f(x) = y$. Essentially, the range of our function $f$ is equal to the codomain. I know that…
user174502
  • 41
4
votes
3 answers

Show that two expressions are equivalent

I am trying to prove a hyperbolic trigonometric identity and I ran into the following expression: $$\frac{\left (\sqrt{x^2+1}+x \right )^2+1}{2\left ( \sqrt{x^2+1} + x \right )} \quad.$$ This expression is supposed to be equivalent…
user4010936
  • 75
4
votes
1 answer

Show that the equation $a_1e^{\alpha_1x} + a_2e^{\alpha_2x} + \cdots + a_ne^{\alpha_nx} = 0$ has at most $n - 1$ real roots.

For non-zero $a_1, a_2, \ldots , a_n$ and for $\alpha_1, \alpha_2, \ldots , \alpha_n$ such that $\alpha_i \neq \alpha_j$ for $i \neq j$, show that the equation $$a_1e^{\alpha_1x} + a_2e^{\alpha_2x} + \cdots + a_ne^{\alpha_nx} = 0$$ has at most $n -…
sourish
  • 598
  • 3
  • 11
4
votes
1 answer

Solving $\frac{\log(x)}{x}=c$, where $c < e^{-1}$

I am just wondering if there is an easy way to solve $$\frac{\log(x)}{x}=c, \text{where } x > 1 \text{ and } c < e^{-1}$$
Elrond Gimli
  • 183
4
votes
2 answers

The juice-problem

I have got an exercise. 7 men are in a room, sitting at a table. In front of each people is a cup. In some cups are juice. The cups altogether have got 3 liters juice in them. Now the first man pour the juice in his cup equally in the other 6 cups.…
Daifus
  • 111
  • 2
4
votes
6 answers

Why is $0/0$ not $\Bbb R$?

Whenever I asked my grade school and college teachers why $\frac00$ is undefined, they would show me a graph of $\frac1x$ and point out the vertical asymptote at $0$, noting that "Since it approaches $-\infty$ from the left and $\infty$ on the…
Ky -
  • 1,300
4
votes
5 answers

Factoring $(x+1)(x+2)(x+3)(x+4)+1$

I have to factor this polynomial: $$(x+1)(x+2)(x+3)(x+4)+1$$ WolframAlpha gives $$(x^2+5x+5)^2$$ How can I prove it without expanding the result? Thanks!
e-man
  • 83
  • 5
4
votes
3 answers

How do you solve for x in this equation?

I tried expanding, but I still can't get rid of the exponents to isolate x. $$\frac{(1+x)^4-1}{x}=4.374616$$ Thank you in advance for your help.
user170492
  • 43
4
votes
1 answer

Why is this answer wrong? (rational expressions)

Simplify the following expression: $$q=\frac { z+9 }{ 5 } +10$$ this is what I got: $$q=\frac { z+9 }{ 5 } +\frac { 50 }{ 5 } $$ $$ q=\frac { z+9+50 }{ 5 } $$ This answer is wrong: $$q=\frac { z+59 }{ 5 } $$
Cherry_Developer
  • 4,239
4
votes
3 answers

How many solutions are there for this equation: $(x^2-x-1)^{x^2}=(x^2-x-1)$

My books says the possible solutions to $\hspace{0.2cm}$$(x^2-x-1)^{x^2}=(x^2-x-1)$ $\hspace{0.2cm}$in $\hspace{0.1cm}$$\mathbb{R}$ $\hspace{0.1cm}$ are $\hspace{0.1cm}$ $-1,1,2$ Is not $\hspace{0.2cm}$ $\frac{1+\sqrt5}{2}$ $\hspace{0.2cm}$ also a…
pirsquare
  • 721
4
votes
2 answers

Logarithm Expansion Question

How do you expand the following logarithm: $$ \log_5 \left(\frac{u}{v^3}\right)^6 $$ The result I got was: $$ 6\log_5u -18\log_5v $$ Is that fully expanded?
user169562
1 2 3
…
99 100