3
votes
2answers
133 views

Combination of quadratic and arithmetic series

Problem: Calculate $\dfrac{1^2+2^2+3^2+4^2+\cdots+23333330^2}{1+2+3+4+\cdots+23333330}$. Attempt: I know the denominator is arithmetic series and equals ...
0
votes
1answer
31 views

Why is $\left(1-\frac{1}{k}\right)^t < e^{-t/k}$?

I came across this statement, but can't see why it holds: $\left(1-\frac{1}{k}\right)^t < e^{-t/k}$ I'm sure it's something simple, but I don't have a great deal of mathematical experience. I ...
3
votes
3answers
227 views

Which is greater: $1000^{1000}$ or $1001^{999}$

Question: Find the greater number: $1000^{1000}$ or $1001^{999}$ My Attempt: I know that: $(a+b)^n \geq a^n + a^{n-1}bn$. Thus, $(1+999)^{1000} \geq 999001$ And $(1+1000)^{999} \geq ...
0
votes
2answers
54 views

Is there a simple algorithm for exponentiating large numbers to large powers?

I've been thinking about this for some days, a multiplication is a lot of sums, so: $$75\times 75=\overbrace{75+75+75+75+75+75+75+75+\cdots}^{\text{75 times}}$$ But then, there is a simple algorithm ...
0
votes
1answer
63 views

Is $\left((-1)^2\right)^\frac12 = (-1)^\left(2\cdot\frac12\right)$? [duplicate]

I'm feeling confused. If I square 1 and -1, the answers should be equal: $1^2 = (-1)^2$ Then I take both sides to the power of $\frac12$: $\left(1^2\right)^\frac12 = \left((-1)^2\right)^\frac12$ ...
0
votes
2answers
73 views

Why is n^(1/m) no valid way to calculate a root

So I came across a situation where a calculator only had square root, but I needed the cubic root. So I used the old $n^\frac13$ trick, and sure enough, the cubic root of n. So this got me thinking. ...
0
votes
3answers
112 views

Find $x ^{ 2013} + 2013x ^{ 2010}$

Q. If $\large {\space x^2 + x + 1 = 0\space } $, Find $ x ^{ 2013} + 2013x ^{ 2010}$. I have tried finding the roots of $x$ from the given equation but that does not work.
6
votes
1answer
201 views

What does the raised $^2$ stand for?

What does the raised $2$ stand for? My first guess was: $4^2$ is $2\times 4=8$? Note: Am not really good at math
0
votes
4answers
49 views

Is the power of 1/2 same thing as principal square root?

$\sqrt{9} = 3$ 9 has 2 square roots: 3 and -3. What is $9^\frac12$? Is $9^\frac12 = \sqrt{9} = 3$ or is $9^\frac12 = \pm3$?
1
vote
4answers
148 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
0
votes
1answer
34 views

Fractional exponentiation in modular arithmetic

Does raising a modular expression to a fraction mean anything? For example, $a\,\,mod \,\,N$ raised to $1/b$ where $b>0$. Does this violate the rules of modularity?
1
vote
2answers
60 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x ...
1
vote
2answers
119 views

How can you prove that a value raised to a fraction($\frac{1}{2}$ for example), is $\sqrt{x}$?

I know that if you raise a value to $\frac{1}{2}$ for example, you take the square root, but that is not what I am asking, what I am asking is; what are you actually doing when raising a value to ...
1
vote
1answer
53 views

Two variable integer equation

I have the following equation: $$ p^q(2^{q-1}-1)=9p^7q $$ I need to solve for $p$ and $q$. $p$ and $q$ are integers. I think I could take the case $p=0$ separately and for that one $q$ could be ...
3
votes
1answer
91 views

find value (-2)^-(2)^(-2)

Find the value of $(-2)^{-(2)^{(-2)}}$. Is it 16/8/-8/none? My attempt: $a^{-x}=\frac1{a^x}$, so, $(-2)^{-(2)^{(-2)}}=(-2)^{\frac{-1}{2^2}}=\frac{1}{(-2)^{\frac14}}$. That is, I would pick 'none ...
0
votes
1answer
31 views

How to evaluate $\pm$ operations

When finding the root of a number with an even exponent, $x^y$ becomes $\pm x$. How would this work in a situation such as $a = \sqrt{(5x + 12)^2 + m}$? I know that the result is not $a = \pm 5x + ...
1
vote
2answers
71 views

Math question from the GMATprep

If $xy=1$ what is the value of: $2^{(x+y)^2}/2^{(x-y)^2}$ A 1 B 2 C 4 D 16 E 19 $(x+y)^2/(x-y)^2$ because $2$ just cancels out from numerator and denominator, ...
0
votes
2answers
139 views

Rational exponents: prove some states

In some rational exponent expressions the solution isn't a real number why? Example (explain what I mean): $$\begin{align} \Big(-x\Big)^{1/n}=\left\{\text{is not a real number}\right\} \end{align}$$ ...
-1
votes
2answers
69 views

Why is $0^0=1$, given the following information? [duplicate]

Why is $0^0=1$, given the following information? We really have two separate rules that are at odds with each other. Typically we have $0^n=0$ (provided n is positive) and $a^0=1$. Each of these ...
0
votes
2answers
69 views

How to find the remainder of $(2010^{1020} + 1020^{2010})$ divided by $3$

What is the remainder when $2010^{1020} + 1020^{2010}$ is divided by 3?
3
votes
2answers
64 views

Proving that not defined value is equal to something

My younger brother (9th Grader) got the following maths problem- Given: $$2^a = 3^b = 6^c$$ Prove: $$c=\frac{a * b}{a+b}$$ From my elementary knowledge of mathematics it seems like a=b=c=0.Also, ...
5
votes
6answers
292 views

How can $4^x = 4^{400}+4^{400}+4^{400}+4^{400}$ have the solution $x=401$?

How can $4^x = 4^{400} + 4^{400} + 4^{400} + 4^{400}$ have the solution $x = 401$? Can someone explain to me how this works in a simple way?
3
votes
2answers
102 views

Division with negative exponents

I have a problem that looks like this: $$\frac{20x^5y^3}{5x^2y^{-4}}$$ Now they said that the "rule" is that when dividing exponents, you bring them on top as a negative like this: ...
10
votes
3answers
832 views

What's the difference between $3^{3^{3^3}}$ and $27^{27}\;$?

Why does $\;\large3^{3^{3^3}}\;$ evaluate to a larger number than $\;\large 27^{27}$?
1
vote
1answer
107 views
2
votes
5answers
274 views

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k$?

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k\;$? I'm aware that $3 = 3^1$ but I would expect $3\cdot 3^k\;$ to be $\;9^k$ or $\;9^{k+1}$.
5
votes
7answers
464 views

Is $0^0=1$ postulate independent of all other axioms of complex numbers?

This question is inspired by the other question which asked for a proof that $i^i$ is a real number. Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
0
votes
2answers
151 views

Why does $(10^4 - 10^2) \cdot 0.0012121212\dots = 12$?

When you answer this question $(10^4 - 10^2) \cdot 0.0012121212\dots$ you get $12$. However, that seems to defy PEMDAS. Please explain. Doing PEMDAS wouldn't you get $(10^4 - 10^2)$ = $10^2$ and then ...
1
vote
0answers
101 views

Collaborative modular exponentiation

EDIT: Rephrased. I have, stored somewhere, the values $a$ , $Q$, $N_1$ (plus its factor) and $a^{2Q} \mod N_1$. I also know $b$, $R$ and $N_2$ (but not its factors). I want to know whether there is ...
66
votes
5answers
2k views

Root Calculation by Hand

Is it possible to calculate and find the solution of $ \; \large{105^{1/5}} \; $ without using a calculator? Could someone show me how to do that, please? Well, when I use a Casio scientific ...
1
vote
1answer
126 views

Convert natural exponent, $e^{c\cdot x}$, into the form $a^{x}$

How does one convert a natural exponent written as $e^{c\cdot x}$ into the form $a^{x}$ ?
0
votes
2answers
155 views

How to define $(-1)^{\frac24}$? [duplicate]

Possible Duplicate: Which step in this process allows me to erroneously conclude that $i = 1$ According to the definition of exponentials, $\displaystyle(-1)^{\frac{2}{4}}$ is equivalent to ...
4
votes
5answers
282 views

What is the remainder of $(14^{2010}+1) \div 6$?

What is the remainder of $(14^{2010}+1) \div 6$? Someone showed me a way to do this by finding a pattern, i.e.: $14^1\div6$ has remainder 2 $14^2\div6$ has remainder 4 $14^3\div6$ has remainder 2 ...
10
votes
5answers
405 views

Why is the math for negative exponents so?

This is what we are taught: $$5^{-2} = \left({\frac{1}{5}}\right)^{2}$$ but I don't understand why we take the inverse of the base when we have a negative exponent. Can anyone explain why?
0
votes
1answer
62 views

Which loan type is cheapest?

I have a 4% loan that spans 20 years, where I pay a fixed amount every three months. If I make an extra payment, I then can choose between two options keep the duration of the loan constant, and I ...
5
votes
2answers
2k views

How can I calculate non-integer exponents?

I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function: $$ f(x,y) = \begin {cases} 1 & y = 0 \\ (x)f(x, y-1) & y > 0 \end ...
2
votes
4answers
219 views

Find largest integer $x$ such that $3^x$ is a factor of $27^5$

Is the following solvable using just arithmetic rather than a calculator, and if so, how? Which of the following numbers is the greatest positive integer x such that $3^x$ is a factor of $27^5~$? ...
1
vote
2answers
400 views

Missing exponent?

$\frac{256}{2^S}=64$ How would you solve for S?
2
votes
5answers
1k views

Why is two to the power of zero equal to binary one?

Probably a simple question and possibly not asked very well. What I want to know is.. In binary, a decimal value of 1 is also 1. It can be expressed as $x = 1 \times 2^0$ Question: Why is two to ...