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.

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194 votes
14 answers

Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\\\\ \frac1{\sqrt{-1}} &= \frac1i \\\\ \frac{\sqrt1}{\sqrt{-1}} &...
Wilhelm's user avatar
  • 2,143
357 votes
31 answers

Is it true that $0.999999999\ldots=1$?

I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is?
134 votes
7 answers

Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$

Why does the following hold: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*} Can we generalize the above to $\displaystyle \sum_{n=...
68 votes
16 answers

Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction

How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
373 votes
23 answers

Zero to the zero power – is $0^0=1$?

Could someone provide me with a good explanation of why $0^0=1$? My train of thought: $$x>0\\ 0^x=0^{x-0}=\frac{0^x}{0^0}$$ so $$0^0=\frac{0^x}{0^x}=\,?$$ Possible answers: $0^0\cdot0^x=1\cdot0^0$,...
Stas's user avatar
  • 3,989
141 votes
36 answers

Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$

Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
b1_'s user avatar
  • 1,585
73 votes
9 answers

Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce abstract ...
187 votes
28 answers

Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$ using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation ...
Fernando Martin's user avatar
144 votes
40 answers

Why is negative times negative = positive?

Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like this: ...
Sev's user avatar
  • 2,043
134 votes
17 answers

Division by zero

I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions. $\dfrac {x}{0}$ is Impossible ( If it's impossible it can't have neither infinite solutions or ...
danielsyn's user avatar
  • 1,577
18 votes
10 answers

How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?

I am trying to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here, so I am ...
Quixotic's user avatar
  • 22.4k
17 votes
4 answers

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant $r$ and how it is derived. For example, when $r = 2$, the formula is given by:$$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1),$$...
hollow7's user avatar
  • 2,455
13 votes
5 answers

Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$

$$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$ I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then ...
seeker's user avatar
  • 6,837
68 votes
6 answers

Strategies to denest nested radicals $\sqrt{a+b\sqrt{c}}$

I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$....
JSCB's user avatar
  • 13.4k
73 votes
22 answers

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
Ssegawa Victor's user avatar
44 votes
9 answers

Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$? [duplicate]

We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$ EDIT: I see this has been dealt with before but at least with ...
Greg's user avatar
  • 575
28 votes
8 answers

How to solve $x^3=-1$?

How to solve $x^3=-1$? I got following: $x^3=-1$ $x=(-1)^{\frac{1}{3}}$ $x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
user avatar
295 votes
22 answers

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
idonno's user avatar
  • 3,889
7 votes
4 answers

Quadratic substitution question: applying substitution $p=x+\frac1x$ to $2x^4+x^3-6x^2+x+2=0$

By using the substitution $p=x+\frac{1}{x}$, show that the equation $$2x^4+x^3-6x^2+x+2=0$$ reduces to $2p^2+p-10=0$. I can't think of anything that produces a useful result, I tried writing p as $p=\...
salman's user avatar
  • 1,680
42 votes
11 answers

Why can a quadratic equation have only 2 roots?

It is commonly known that the quadratic equation $ax^2+bx+c=0$ has two solutions given by: $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ But how do I prove that another root couldn't exist? I think ...
Atul Mishra's user avatar
  • 3,096
8 votes
5 answers

Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd

I read somewhere that $(a^n - b^n)$ It is always divisible by $a-b$. When $n$ is even it is also divisible by $a+b$. When $n$ is odd it is not divisible by $a+b$. and $(a^n + b^n)...
aarbee's user avatar
  • 8,108
21 votes
5 answers

Why $(-2)^{2.5}$ isn't equal to $((-2)^{25})^{1/10}?\,$ [Fractional powers of negative numbers]

I've tried both calculations on Wolfram Alpha and it returns different results, but I can't get a grasp of why it is like that. From my point of view, both calculations should be the same, as $2.5=25/...
Rizescu's user avatar
  • 313
171 votes
15 answers

Is there a general formula for solving Quartic (Degree $4$) equations?

There is a general formula for solving quadratic equations, namely the Quadratic Formula, or the Sridharacharya Formula: $$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } $$ For cubic equations of the ...
John Gietzen's user avatar
  • 3,451
29 votes
5 answers

When do we get extraneous roots?

There are only two situations that I am aware of that give rise to extraneous roots, namely, the “square both sides” situation (in order to eliminate a square root symbol), and the “half absolute ...
Mike Jones's user avatar
  • 4,450
56 votes
11 answers

Is $\sqrt{64}$ considered $8$? or is it $8,-8$?

Last year in Pre-Algebra we learned about square roots. I was taught then that $\sqrt{64}=8$ and $\sqrt{100}=10$, which I understood and accepted. I was also taught that $\pm\sqrt{64} = 8,-8$ because ...
Nico A's user avatar
  • 4,934
24 votes
5 answers

Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$

I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a ...
JSchlather's user avatar
  • 15.4k
2 votes
9 answers

How to factor $\,9x^2-80x-9?\,$ (AC-method) [closed]

Is there a general method to factor a quadratic like $9x^2-80x-9?\,$ I'm having a lot of difficulty due to the leading coefficient being unequal to $1$?
user3131263's user avatar
99 votes
15 answers

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
newcomer's user avatar
  • 963
52 votes
4 answers

Purely "algebraic" proof of Young's Inequality

Young's inequality states that if $a, b \geq 0$, $p, q > 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$). Back when I ...
asmeurer's user avatar
  • 9,666
35 votes
3 answers

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+\cdots+\tan^2 89^\circ = \text{?}$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+\cdots+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch ...
Ninja Boy's user avatar
  • 3,103
16 votes
6 answers

Simplifying $\sqrt[4]{161-72 \sqrt{5}}$

$$\sqrt[4]{161-72 \sqrt{5}}$$ I tried to solve this as follows: the resultant will be in the form of $a+b\sqrt{5}$ since 5 is a prime and has no other factors other than 1 and itself. Taking this ...
1110101001's user avatar
  • 4,118
465 votes
10 answers

Is this Batman equation for real? [closed]

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? Batman Equation in text form: \begin{align} &\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-...
a_hardin's user avatar
  • 5,491
15 votes
5 answers

Given that $x^y=y^x$, what could $x$ and $y$ be?

It's not too difficult to figure out that $x$ and $y$ can both be 1, and also $x$ can be 2 and $y$ can be 4 (and vice versa). But I can't rule out if there are other solutions. Does it have anything ...
yroc's user avatar
  • 1,035
10 votes
5 answers

Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$

By considering: $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^1}{n^{2}} = \frac 1 2$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^2}{n^{3}} = \frac 1 3$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^3}{n^{4}} = \...
JSCB's user avatar
  • 13.4k
223 votes
10 answers

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
David G's user avatar
  • 4,177
5 votes
6 answers

Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case]

In respect to a larger proof I need to prove that $(n-m) \mid (n^r - m^r) $ (where $\mid$ means divides, i.e., $a \mid b$ means that $b$ modulus $a$ = $0$). I have played around with this for a while ...
Callum Rogers's user avatar
61 votes
12 answers

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
Daniel R. Collins's user avatar
12 votes
5 answers

Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$

I am having trouble with this problem from my latest homework. Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ ...
Steve's user avatar
  • 155
39 votes
4 answers

Integration by partial fractions; how and why does it work?

Could someone take me through the steps of decomposing $$\frac{2x^2+11x}{x^2+11x+30}$$ into partial fractions? More generally, how does one use partial fractions to compute integrals $$\int\frac{P(...
Finzz's user avatar
  • 1,069
162 votes
19 answers

What actually is a polynomial?

I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial. I wasn't in the advanced mathematics class in 8th grade, then in 9th ...
Travis's user avatar
  • 3,396
95 votes
17 answers

What is the most elegant proof of the Pythagorean theorem? [closed]

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
41 votes
5 answers

How to find the sum of the series $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$?

How to find the sum of the following series? $$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}$$ This is a harmonic progression. So, is the following formula correct? $\frac{(number ~...
Sachin's user avatar
  • 9,686
33 votes
9 answers

Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$

In Spivak's Calculus 3rd Edition, there is an exercise to prove the following: $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$ I can't seem to get the answer. Either I've gone ...
jamesbrewr's user avatar
3 votes
4 answers

Multiples are closed under integral linear combinations

My lecturer gave us the following side note when explaining the euclidean algorithm in class. Eucledian Algorithm: Let $a$ and $b$ be natural numbers, then there are integers $m$ and $n$ such that $\...
Scb's user avatar
  • 260
73 votes
6 answers

$1 + 2 + 4 + 8 + 16 \ldots = -1$ paradox

I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1: Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 \...
Christian's user avatar
  • 841
47 votes
3 answers

Find all real numbers $x$ for which $\frac{8^x+27^x}{12^x+18^x}=\frac76$

Find all real numbers $x$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac76$$ I have tried to fiddle with it as follows: $$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$ $$ 3 \cdot 2^{3x+1}...
John Marty's user avatar
  • 3,650
31 votes
5 answers

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction, I tried with Wolfram Alpha to check the validity, ...
Quixotic's user avatar
  • 22.4k
16 votes
11 answers

$\tan A + \tan B + \tan C = \tan A\tan B\tan C\,$ in a triangle

I want to prove this (where each angle may be negative or greater than $180^\circ$): When $A+B+C = 180^\circ$ \begin{equation*} \tan A + \tan B + \tan C = \tan A\:\tan B\:\tan C. \end{equation*} We ...
trig's user avatar
  • 169
101 votes
14 answers

Why can't you square both sides of an equation?

Why can't you square both sides of an equation? I've been asked this many times and can never quite give a good, clear, concise answer (for beginning algebra students) in plain language. I just ...
Jeff's user avatar
  • 3,345
40 votes
7 answers

prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer

Prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer. My thought process: The numerator is the product of the first $n$ even numbers and the product of the first $n$ odd numbers; ...
hdtv1104's user avatar
  • 601

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