Questions about exponentiation

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If you rewrite (3^12)(3^-12) in the form 3^n what does it equal? and what is the intuition behind it?

Do exponents cancel each other out so it is just 3? or do the negatives cancel out (-x*-x=x) so it is 3^24? or is it something else?
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2answers
21 views

derivative of $e^{\ln x^2}-3x^7$

$$e^{\ln x^2}-3x^7$$ The first term: $=e^v$ $v=\ln x^2=u^2$ $v\;'=2uu\;'=(2\ln x)\dfrac{1}{x}=\dfrac{2\ln x}{x}$ $\dfrac{e^{\ln x^2}2\ln x}{x} +21x^{-8}$ How do I simplify further? I don't ...
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0answers
18 views

How to make powers work right on a casio calculator?

I pressed some random buttons and it worked fine,but now its not working again. for example my calculator shows that -6^2 is - 36 and not 36, anyone know how to fix it??? My clculator model is casio ...
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3answers
28 views

Derivative of $e^\sqrt{4x+4}$

$$f(x)=e^\sqrt{4x+4}$$ $f(x)=e^u$ $u=\sqrt{4x+4}=(4x+4)^{1/2}$ $u\;'=\dfrac{1}{2}(4x+4)^{-1/2}=\dfrac{1}{2\sqrt{4x+4}}$ I don't know how to proceed from here. Thanks.
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1answer
29 views

The sum of powers of two

While doing a proof of correctness for an algorithm, I ran into a roadblock with a smaller proof. The problem in words: A set with all elements which are a power of 2, and the sum of the set is ...
2
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3answers
20 views

Calculating sum of consecutive powers of a number

Here is my problem, I want to compute the $$\sum_{i=0}^n P^i : P\in ℤ_{>1}$$ I know I can implement it using an easy recursive function, but since I want to use the formula in a spreadsheet, is ...
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1answer
32 views

What is the value of x? Related to Indices.

Just some days ago I appeared for a maths exam. In that exam there was a question related to Indices which I was not able to solve. After the exam I even tried solving it in the home next 2 days but ...
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2answers
37 views

Evaluating a limit as $x \to -\infty$ of a power of a rational function

Sorry for the weird title, I don't know how to put the equation on the title. $$\lim_{x\to-\infty}\left(\frac{1-x^3}{x^2+7x}\right)^5$$ Ok I divided inside the parenthesis by $x^2$, but now I am ...
-1
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3answers
133 views

Is it true that $0^0$ is undefined? Why or why not? [duplicate]

Is it possible for zero to the power of zero to be undefined? Is there a good reason if it IS undefined? If yes, I hope there is! This question is different because I'm trying to figure out if $0^0$ ...
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1answer
45 views

Right way to solve for $\frac{2^{900}*7^{898}}{14^{897}}$

As a sequel to my question How to solve $0.5^{1200}\times (2^{1204})$? : $\frac{2^{900}*7^{898}}{14^{897}}$ Will I first solve the upper raw like did in previous question and then anwser $14^{897}$
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4answers
77 views

How to solve $0.5^{1200}\times (2^{1204})$?

I've been struggling with this one. I know that the anwser is $16$, but how do I solve this on paper? $0.5^{1200}\times 2^{1204}$ I know that this has something to do with first subtracting the ...
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3answers
44 views

Calculating the value with large exponents

I'm trying to solve $2014^{2015}$ $\pmod {11}$, is there a trick or tip to break the problem down to make it easier to solve?
-1
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3answers
38 views

Why is $3^{(x-5)} + 3^{(x-7)} + 3^{(x-9)} = 91$?

So far I think that this is somehow related to that $(x-7) - (x-5) = (x-9) - (x-7) = 2$, but is it ? What steps do you take to add $3^{x-5} + 3^{x-7} + 3^{x-9}$ up ? Thank you!
2
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2answers
26 views

Chaining Exponent Rules Together

I'm having trouble understanding why the following property is true and want to make sense of it before going ahead and use it in my proof by induction: $$2^{2^n}=2^{2^{n-1}}\times 2^{2^{n-1}}=\left( ...
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1answer
20 views

Power function and involution $f(x) = x^a$

For power functions we have a variable $x$ and a constant $a$; we get that $f(x) = x^a$. Find all involutions for $f(x)$. I started out with basic functions such as $f_1(x) = f_1^{-1}(x) = x^1$ and ...
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3answers
74 views

Is there a shortcut for raising 2 to the power of a number (e.g. $2^{27}$)?

In networking, when dealing with subnetting, you convert the net mask to binary and count the number of ones (for the example in the question there would be $27$ $1$'s) and to figure out how many ...
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3answers
62 views

How can we prove that the square root of any number is equal to the statment given below

Is there any theorem that can explain this $\sqrt[n]{x}= x^{1/n} $, is there any practical example of $ x^{1/n} $, 1/n times of a number.
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0answers
15 views

Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it ...
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1answer
9 views

Confusion in “mask bits”

This is the question: By definition of the IEEE754 standard, 32-bit floating point numbers are represented as follows: S (1 bit) E (8 bits) M (23 bits) S: Sign bit E: Exponent M: ...
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2answers
26 views

Simplify an expression involving indices

One of my friends asked me this question: Simplify $$\frac{50^{3x-1} 10^{2-3x}}{250^{3x+1}}$$ I've been thinking about the question for more than a day. I've looked through my teacher's notes but ...
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2answers
43 views

Solving equations with fractional indices

How would I go about solving an equation like this? $3^{4/3}b^{5/3} - b^3 = 1$ I thought about rearranging to get $3^{4/5}b = (1 + b^3)^{3/5}$, but that didn't seem to lead anywhere as I couldn't ...
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2answers
17 views

Exponents with the power being a negative/decimal?

How would I do the equation $b^{n}$ Or $b^{-n}$ Where $b$ is the base, and $n$ in a negative/decimal?
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6answers
2k views

What's the inverse operation of exponents?

You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the ...
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1answer
41 views

Irrational power of root

Let $a$ and $b$ be rational numbers, such that $\sqrt{a}$ and $\sqrt{b}$ are irrational. Can $\sqrt{a}^\sqrt{b}$ be rational? I found examples, where the irrational power of an irrational number is ...
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1answer
71 views

Simplifying $2^\sqrt{\log x}$

Can this expression be simplified? $$2^\sqrt{\log x}$$ Thank you
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2answers
36 views

Power of very big numbers

So, I was solving a question, and I came across this. If I have, x=a^b, and If I want to calculate the last digit of x, then it ...
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4answers
72 views

$4^{x}+2^{x+1}=18$ Please help me solve?

I tried using logs on both sides or tried treating it as a quadratic but didn't manage to simplify it, Help?:D
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2answers
80 views

I need to solve this equation [do my homework]. [closed]

Problem for my university math class 1st semester Thanks a lot!
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1answer
15 views

exponential equation (solve for x)

We started review in my calculus class and I have mostly forgotten everything about exponential equations. $$ e^{-x} * (3e^{2x}-(5/4))^{1/2} = e^x $$ Would x just equal 0, when -x+x=0 (two sides of ...
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1answer
64 views

is there any solution for $x^2 +x + 2 = e^x$ by using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
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3answers
122 views

Matrix exponential: Formal notation for power series? Or, more?

For a square matrix $A$, I'm already used to see and use: $$\sum_{n=0}^{\infty} \frac{A^n}{n!} = \lim_{n \to \infty} \left(I + \frac{A}{n}\right)^n = e^A$$ Which means a matrix $A$ is just like some ...
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2answers
28 views

Negative Number raised to fractional power

How would you solve a negative number raised to a fraction a/b if b is odd and a is evem? Ignoring imaginary numbers i.e $(-1)^\frac23$ Calculator returns an error $(-1)^\frac 13 (-1)^\frac 13$ = ...
0
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2answers
68 views

Can $x^2 +x + 2 = 10^x$ be solved using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
1
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2answers
28 views

Exponential equations with variables on both sides

I have the following: $$8^{3x+4} = 5^{4x-2}$$ How would I solve this? I tried this: $$(3x+4)\log 8 = (4x-2)\log 5$$ but have no idea where to go from there. Thank you!
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1answer
15 views

calculating a decrease at constant instantaneous rate

If we have a value e.g. $$ B = 20000 $$ and it decreases at a constant instantaneous rate of say $$ -1.1*10^{-2} $$ per unit time. What would B look like over say 300 time units, and how do we ...
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1answer
36 views

How to find the common base of terms in an expression?

I'm teaching myself basic algebra from a book and am stuck on a question. In the current section it is about expressing numbers as powers of the same base. So $9$ maybe expressed as $3^2$. Another ...
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1answer
18 views

Plotting nth root of x against nx on a graph.

I've spent the last two days trying to figure this out. What I'm trying to do is rearrange this: $$ x^n = \frac xn $$ to make n the subject, to allow me to plot on a graph, with n being the ...
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1answer
23 views

Sign of Fractional Exponents

When calculating a number to a fractional exponent or fractional nth root, in what cases is there both a positive and negative solution as opposed to just a positive or just a negative solution?
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1answer
75 views

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x … $ are all integers then $x$ is an integer as well?

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
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1answer
31 views

exponentials with different base

What rule can I use when solving exponentials like this one $\frac {2^6 \cdot 5^8 \cdot 3}{100^3}$ I know how to solve exponentials when the bas number is the same with these formulas $x^m \cdot x^n ...
2
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2answers
47 views

Unit Quaternion to a Scalar Power

I'm trying to modify a physics engine for efficiency. Currently, as objects move around the world, their orientation (a quaternion) is updated every frame, by multiplying by the rotation (another ...
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0answers
53 views

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
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1answer
30 views

From an expression raised in a power of 2 to an expression raised in the power or 10

Is there a simple/"easy" way to convert a big number from a power of $2$ to a power of $10$ equivalent. Example: I had $2^{127}\cdot 1.9999999$ which I did the multiplication got the result and from ...
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4answers
43 views

Why is a Constant added to front?

I made the differential equation : $$dQ = (-1/100)2Q dt$$ I separate it and get: $\int_a^b x (dQ/Q) = \int_a^b x (-2/100)dt$ this leads me to: $\log(|Q|) = (-t/50) + C$ I simplify that to $Q = ...
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1answer
42 views

Big-Oh of exponent of exponent

How does one whether an exponent of an exponent is the big-Oh of the other? For example, if I have $a^{b^n}$ and $b^{a^n}$, how would i determine and prove which is a big oh of another? I'm thinking ...
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4answers
73 views

Existence of solution in $x,y \in (a,b)$ of $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$

Let $a<b$ be positive real numbers , then is it true that there exist $x,y \in (a,b)$ such that $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$ ?
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1answer
31 views

Methods to calculate powers in my head

How can i calculate powers in my head, like small powers. for example $0.5^3$, how can i work this out quickly and easily? or $4^5$
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1answer
54 views

Definition of $z^0, z=a+bi, a,b \in \mathbb{R}, z \neq 0$

Today I found in a True-False the question; Does the equality $$z^0=1, z=a+bi, a,b \in \mathbb{R}$$ hold $\forall z \in \mathbb{C^*}$? The thing is, this was never clearly defined in the book, and ...
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0answers
25 views

Iterations $n, n^n, (n^n)^{(n^n)},…$

(Note: I'm reposting this, as I posted the original too late in the evening to gain anyone's notice.) A contest problem (#2 on the 2010 Virginia Tech Math Competition) proffers the solver the ...
3
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2answers
91 views

Solve the equation: $1+2^x+4^x+8^x+16^x+32^x=3(1+2^x+4^x)$

I am doing some math repetition and am a bit stuck on this exercise: Solve the equation: $1+2^x+4^x+8^x+16^x+32^x=3(1+2^x+4^x)$. Now, this is a geometric sum on both the $LHS$ and $RHS$, which I ...