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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Real solutions of $x$ in $\{x\} = \{x^2\} = \{x^3\}$

Calculate real values of $x$ in $\{x\} = \{x^2\} = \{x^3\}$ , where $\{x\}$ is the fractional part of $x$. My Attempt: Let $\{x\} = \{x^2\} = \{x^3\} = k$. Because the fractional part of $X$ is given by $\{X\} = X-\lfloor X \rfloor$, we know the…
juantheron
  • 53,015
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Prove that $\log_{2}(7)$ is irrational

Prove that $\log_{2}(7)$ is an irrational number. My Attempt: Assume that $\log_{2}(7)$ is a rational number. Then we can express $\log_{2}(7)$ as $\frac{p}{q}$, where $p,q\in \mathbb{Z}$ and $q\neq 0$. This implies that $7^q = 2^p$, where either…
juantheron
  • 53,015
9
votes
3 answers

If $x^2-16\sqrt x =12$ what is the value of $x-2\sqrt x$?

I saw this problem:If $x^2-16\sqrt x =12$ what is the value of $x-2\sqrt x$? To make notations simpler Let $x=t^2$ and $t^4-16t=12$ and $l= t^2-2t$. I tried to use the fact the $\frac{12} l =\frac{t^4-16t}{t^2-2t}= t^2+2t +4 + \frac{8}{t-2}= (t-2)^2…
pie
  • 4,192
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3 answers

If $x^2=y^2$, prove that $x=y$ or $x=-y$

I have a simple question here. I am trying to prove that, given $x^2=y^2$, $x=y$ or $x=-y$. I know exactly why this is true; it's obvious. I'm just unclear on the general format of a proof, as well as how I should specifically write this one. Any…
Zane Victor Kaminski
  • 253
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What are some common pitfalls when squaring both sides of an equation?

Squaring both sides of the equation $x=-\sqrt{y+1}$ leads to an equation that has more solutions than the original equation. When can we be sure that squaring both sides will result in the same solution set? What operations on both sides are…
hondaman
  • 565
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The Product Rule of Square Roots with Negative Numbers

In the statement $\forall a, b \geq0, \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, why is it necessary to restrict $a$ and $b$ to being $\geq 0$? It seems that one should be able to say, for example, $(-3)^{1/2} \cdot (-3)^{1/2} = (-3 \cdot -3)^{1/2} =…
anse
  • 93
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Why can I square both sides?

I am not used to English. I ask for your understanding in advance. There is the equation: $ x= 2^\frac{1}{2}$ we can square both side like this: $ x^2= 2$ But I don't understand why that it's okay to square both sides. What I learned is that adding,…
KimNoob
  • 101
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Solving a algebraic equation

I am to solve the following equation third order equation $x^3+x^2+x+1=0$ What I've tried so far is writing the equation as $ x \cdot (x^2+x+1)+1=0$ but that didn't lead anywhere. How do I solve this without using a computer?
chase.furlong
  • 91
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Is there a way to show $\frac{1646-736\sqrt{5}}{2641-1181\sqrt{5}}=\frac{17+15\sqrt{5}}{7+15\sqrt{5}}$ without multiplying large numbers?

I need to show that the following equality holds $$\frac{1646-736\sqrt{5}} {2641-1181\sqrt{5}} =\frac{17+15\sqrt{5}} {7+15\sqrt{5}} $$ The only way I could prove it was via cross multiplication using a calculator. Is there any simpler way to do…
Paramanand Singh
  • 87,309
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4 answers

Algebra problem (problem from Swedish 12th grade ‘Student Exam’ from 1932)

The following problem is taken from a Swedish 12th grade ‘Student Exam’ from 1932. The sum of two numbers are $a$, the sum of the 3rd powers is $10a^3$. Calculate the sum of the 4th powers, expressed in $a$. Is there a shorter/simpler solution…
mf67
  • 853
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4 answers

finding the real values of $x$ such that : $x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$

How to find the real values of $x$ such that : $$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
user67465
  • 93
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2 answers

How to solve $4^x - 2^{x + 1} = 3 $ for x?

We figured that this can be changed to $2^{2x} - 2^x \cdot 2 = 3$, but couldn't solve from there. Perhaps we are not on the right path?
user1729843
  • 91
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7 answers

How to simplify a square root

How can the following: $$ \sqrt{27-10\sqrt{2}} $$ Be simplified to: $$ 5 - \sqrt{2} $$ Thanks
user66659
  • 155
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2 answers

Domain of composition of functions

I'm teaching a precalculus course and was wanting to let my students try to solve the following problem. If $$ f(x)=\sqrt{x}, g(x)=\frac{x}{x-1},h(x)=\sqrt[3]{x} $$ Find the domain of $$f\circ g\circ h $$ We have the following. $$ (f\circ g\circ…
JohnC
  • 361
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2 answers

Raising both sides of an equation to the same negative power

If I raise both sides of an equation to the same negative exponent will the equality remain? $$a = b$$
Omicron
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