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.
4,051
questions
194
votes
14
answers
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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}} &...
- 2,143
357
votes
31
answers
59k
views
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
96k
views
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
52k
views
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
51k
views
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:
$0^0\cdot0^x=1\cdot0^0$,...
- 3,989
141
votes
36
answers
304k
views
Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
- 1,585
73
votes
9
answers
31k
views
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
19k
views
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 ...
- 5,795
144
votes
40
answers
108k
views
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:
...
- 2,043
134
votes
17
answers
13k
views
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 ...
- 1,577
18
votes
10
answers
17k
views
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 ...
- 22.4k
17
votes
4
answers
12k
views
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:$$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1),$$...
- 2,455
13
votes
5
answers
33k
views
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 ...
- 6,837
68
votes
6
answers
21k
views
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 $1-\sqrt{2}$....
- 13.4k
73
votes
22
answers
18k
views
Prove $0! = 1$ from first principles
How can I prove from first principles that $0!$ is equal to $1$?
- 1,007
44
votes
9
answers
10k
views
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 ...
- 575
28
votes
8
answers
33k
views
How to solve $x^3=-1$?
How to solve $x^3=-1$? I got following:
$x^3=-1$
$x=(-1)^{\frac{1}{3}}$
$x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
user2723
295
votes
22
answers
48k
views
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 ...
- 3,889
7
votes
4
answers
1k
views
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=\...
- 1,680
42
votes
11
answers
14k
views
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 ...
- 3,096
8
votes
5
answers
40k
views
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)...
- 8,108
21
votes
5
answers
1k
views
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/...
- 313
171
votes
15
answers
353k
views
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 ...
- 3,451
29
votes
5
answers
22k
views
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 ...
- 4,450
56
votes
11
answers
12k
views
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 ...
- 4,934
24
votes
5
answers
9k
views
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 ...
- 15.4k
2
votes
9
answers
2k
views
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 $1$?
- 45
99
votes
15
answers
8k
views
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:
$$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
- 963
52
votes
4
answers
22k
views
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 ...
- 9,666
35
votes
3
answers
4k
views
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 ...
- 3,103
16
votes
6
answers
14k
views
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 ...
- 4,118
465
votes
10
answers
520k
views
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|-...
- 5,491
15
votes
5
answers
3k
views
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 ...
- 1,035
10
votes
5
answers
2k
views
Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$
By considering:
$$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^1}{n^{2}} = \frac 1 2$$
$$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^2}{n^{3}} = \frac 1 3$$
$$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^3}{n^{4}} = \...
- 13.4k
223
votes
10
answers
13k
views
What does $2^x$ really mean when $x$ is not an integer?
We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
- 4,177
5
votes
6
answers
916
views
Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case]
In respect to a larger proof I need to prove that $(n-m) \mid (n^r - m^r) $ (where $\mid$ means divides, i.e., $a \mid b$ means that $b$ modulus $a$ = $0$). I have played around with this for a while ...
- 185
61
votes
12
answers
17k
views
What are the Laws of Rational Exponents?
On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
- 8,189
12
votes
5
answers
8k
views
Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$
I am having trouble with this problem from my latest homework.
Prove the arithmetic-geometric mean inequality. That is, for two positive real
numbers $x,y$, we have
$$ \sqrt{xy}≤ \frac{x+y}{2} .$$
...
- 155
39
votes
4
answers
13k
views
Integration by partial fractions; how and why does it work?
Could someone take me through the steps of decomposing
$$\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(...
- 1,069
162
votes
19
answers
21k
views
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 ...
- 3,396
95
votes
17
answers
65k
views
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
20k
views
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 ~...
- 9,686
33
votes
9
answers
117k
views
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 ...
- 565
3
votes
4
answers
776
views
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 $\...
- 260
73
votes
6
answers
24k
views
$1 + 2 + 4 + 8 + 16 \ldots = -1$ 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 \...
- 841
47
votes
3
answers
13k
views
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:
$$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$
$$ 3 \cdot 2^{3x+1}...
- 3,650
31
votes
5
answers
2k
views
How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?
Other than the general inductive method,how could we show that $$\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, ...
- 22.4k
16
votes
11
answers
48k
views
$\tan A + \tan B + \tan C = \tan A\tan B\tan C\,$ 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 ...
- 169
101
votes
14
answers
115k
views
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 ...
- 3,345
40
votes
7
answers
29k
views
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; ...
- 601