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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What is the property that allows $5^{2x+1} = 5^2$ to become $2x+1 = 2$?

What is the property that allows $5^{2x+1} = 5^2$ to become $2x+1 = 2$? We just covered this in class, but the teacher didn't explain why we're allowed to do it.
user493941
  • 73
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What is the smallest possible value of $q$ such that $\frac{7}{10}<\frac{p}{q}<\frac{11}{15}$?

If $p$ and $q$ are positive integers such that $\frac{7}{10}<\frac{p}{q}<\frac{11}{15}$ then the smallest possible value of $q$ is: $(A)\quad 60;\quad (B)\quad 30;\quad (C)\quad 25;\quad (D)\quad 7$. What is the correct way to solve this kind of…
keylin.rion
  • 163
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Calculating $e^{ar}+e^{ar^2}+...+e^{ar^n}$

Calculate the sum, $$e^{ar}+e^{ar^2}+...+e^{ar^n}\ \text{where} \ a,r\in \mathbb{R}$$ It's easy to calculate the sum when the powers of $e$ are in an arithmetic progression. How do we proceed when the powers are in geometric progression?
user694028
7
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4 answers

How to solve equations like $8^{2n+1} = 32^{n+1}$

I have stumbled on this question, and there are a few questions after it of the same type. How do I solve it and what is the right approach for this kind of question? $$8^{2n+1} = 32^{n+1}$$ I need to find the value of $^n$. Step by step if at all…
sprocket12
  • 337
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Proving basic math principles?

I've been thinking how to prove some of these basic "formulas" but most of them I don't know how: $$\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$$ exponentials such as $$(a^b)\cdot (a^c)=a^{b+c}$$ (I know this makes sense for integers but how do you…
Ovi
  • 23,737
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Can the domain of $f(x)/g(x)$ be larger than that of $f(x)$ or $g(x)$?

Consider the following two functions: $f(x) = \sqrt {4 - x^2};\;\{x : x \in \mathbb{R}, x \ge -2 \,\ \text{and} \ \, x \le 2\}$ $g(x) = \sqrt {1 + x};\;\{x : x \in \mathbb{R}, x \ge -1\}$ Given that: $$\frac{f(x)}{g(x)} =…
jwool
  • 71
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Why didn't they simplify $x^y=y^x$ to $x=y$?

Solving $x^y = y^x$ analytically in terms of the Lambert $W$ function This "solution" for $x^y=y^x$ should simplify to $y=x$, but for some reason no pointed that out in the OP. According to the stack exchange, the answer is $y=…
user14554
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Long Division Algorithm Proof

As I was doing my homework today, a sudden thought popped into my head. Why does our long-division algorithm work and how can I prove it? Why does it the function the way it does? Why do we not do division starting from the right and going to the…
Dude156
  • 1,316
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Algebra Mess, don't know how to proceed

I have kind of an algebra problem. The original question is a Bilinear transformation for analogue to digital filters. (This is not a homework question) In my lecture notes, he goes to the answer like 1 step, I'm trying to work it out but I'm…
AlfroJang80
  • 253
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Why, given a natural number $n$, does $n^6$ always have the remainder of 1 when divided by 7?

I am trying to prove why a natural number $n$ (which is not a multiple of $7$) when taken to the power of six ($n^6$) and divided by 7 always have the remainder of 1? I am not supposed to use "Fermats little theorem", but I am given the hint that…
Lukas Arvidsson
  • 2,376
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Finding the value of $\frac{13}{a-13}+\frac{23}{b-23}+\frac{42}{c-42} $ given a system of three equations

Let $a, b, c, x, y, z$ be real numbers that satisfy the three equations $$ 13x+by+cz=0 $$ $$ ax+23y+cz=0 $$ $$ ax+by+42z=0 $$ Suppose that $ a\neq13 $ and $x\neq0$. What is the value of $$\frac{13}{a-13}+\frac{23}{b-23}+\frac{42}{c-42} $$ I…
Pizzaroot
  • 453
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Minimum of $\left| \sin x- 1\right| + \left|\sin x- 2\right| + \left| \sin x -3\right| + \left| \sin x+1\right|$

I would like to know the minimum value of $$\left| \sin x- 1\right| + \left|\sin x- 2\right| + \left| \sin x -3\right| + \left| \sin x+1\right|$$ for $x \in \mathbb{R}$.
juantheron
  • 53,015
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Solve in $\mathbb R\quad 5^\sqrt{x} - 5^{x-7} = 100$

Solve in $\mathbb R$ $$5^\sqrt{x} - 5^{x-7} = 100$$ $\mathbf {My Attempt}$ I converted the eq. to this form $$5^{(\sqrt{x}-3)(\sqrt{x}+3)}-5.5^{\sqrt{x}-3}+4=0$$ It's apparent that $\;\mathbf {x=9}\; $ is a solution, but I can't find the reasoning…
Wolfdale
  • 749
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Where is the step that caused the extraneous solution?

$$x^{1/3}+x^{1/6}-2=0\iff (x^{1/6}-1)(x^{1/6}+2)=0$$ $\iff x^{1/6}=1$ or $x^{1/6}=-2$ $\implies x=1^6=1$ or $x=(-2)^6=64$ By checking when $x=64$:$\sqrt[3]{64}+\sqrt[6]{64}-2=4+2-2=4\ne0$ Which step caused the extraneous solution $x=64$ to appear
Zyad Yasser
  • 498
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2 answers

Area between two overlapping triangles

The shaded part for 1 single triangle is $4/9$ths of the total area of the triangle. If this was considered to be 4 units, then the unshaded $5/9$ths would be 5 units. Thus the total area of the whole figure is 14 units and so $4/14$ths or $2/7$ths…
salman
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