What is Exponents: Definition and 268 Discussions

Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n, and pronounced as "b raised to the power of n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:









{\displaystyle b^{n}=\underbrace {b\times \dots \times b} _{n\,{\textrm {times}}}.}
The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the nth power", "b raised to the power of n", "the nth power of b", "b to the nth power", or most briefly as "b to the nth".
One has b1 = b, and, for any positive integers m and n, one has bn ⋅ bm = bn+m. To extend this property to non-positive integer exponents, b0 is defined to be 1, and b−n (with n a positive integer and b not zero) is defined as 1/bn. In particular, b−1 is equal to 1/b, the reciprocal of b.
The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.
Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

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  1. Juanda

    I 10^(1/10) is VERY close to 2^(1/3)

    Is there a subjacent reason that explains why these two numbers are so close? $$10^{1/10} \approx 2^{1/3}$$ For context, this is where I found out about this. Source: https://www.instarengineering.com/pdf/resources/Instar_Vibration_Testing_of_Small_Satellites_Part_5.pdf Is it just a...
  2. G

    Difference between -3² and (-3)² ?

    What is the difference between these to. -3² and (-3)² ? I know - x - = + I am told -3²= -9 not +9 -3² means -3 x -3 = 9 I am told (-3)² = 9
  3. brotherbobby

    Inequality involving a reciprocal - where's the mistake?

    Problem Statement : Solve the inequality : ##\left( \dfrac{1}{3} \right)^x<9##. Attempts: I copy and paste my attempt below using Autodesk Sketchbook##^{\circledR}##. The two attempts are shown in colours black and blue. Issue : On checking, the first attempt in black turns out to be...
  4. mcastillo356

    Pattern of variables with absolute value exponents

    On ##x\in{(-1,1)}##, ##x\in{\mathbb{R}}##, ##\forall{n\in{\mathbb{N}}}##, ##x^{|2n|}=O(x^{|2n+1|})## Sugestions? Any answer is wellcome! Greetings, PF
  5. hackedagainanda

    Exponents clarification question

    So my attempt is this: ##(2k)^2 k^3 = 4k^5## to clear the top numerator then to clear the denominator ## k^{-1} k^{-5} = k^{-6}##Then I apply the quotient rule and get ##4k^{11} (5k^{-2})^{-3}## and simplifying the right hand side I get ##5^{-3} k^6## here is where I got lost, why is it that...
  6. sahilmm15

    B Using Small Examples to Solve Combinatorial Problems

    In the below picture I understood the problem and also understood the solution. But I am not clear on why did they choose this particular method. So my question is why did they divided 50 with powers of 2 and 3, and what relation does ##50## have with ##50!## ( I am clear on the solution part...
  7. A

    Powers with fractional exponents that have an even denominator

    (If I should have posted this in the Math thread instead of the Homework thread, please let me know.) I have three questions which I will ask in sequence. They all relate to each other. I've typed my questions and solutions attempts below. I've also attached a hand-written version of this...
  8. hackedagainanda

    Why do fractional exponents result in the square root operator?

    So I got the answer through a little addition i.e 9^(1/2) multiplied by 9^(1/2) = 9^1 or 9 3 x 3 = 9 so 3 is the answer to what is 9^(1/2) I've tested this out with a few other numbers and have made this generalization, x^(1/2) = √x It seems to make the equations orderly and consistent but is...
  9. e2m2a

    I Prime factorization and real exponents

    I know that the prime factorization theorem predicts that a prime number raised to an integer power will never be equal to another prime number raised to a different power. But does this apply to real number powers? For example, suppose there is a prime number raised to some real value, could...
  10. karush

    MHB Gre Al 3 Exponents | Physics Forums

    https://www.physicsforums.com/attachments/9551 OK I went thru this 3 time and might still have some error ... otherwise typos maybe I don't think this could be solved by just observation
  11. D

    MHB Calculating Exponents for Business Model

    Hi, Everyone. I suspect I am a bit unique here. I'm struggling with a math problem for a business model rather than a homework assignment. It's been quite a while since I have worked with exponents and I am hoping someone can assist me with a question. I have the following calculations...
  12. E

    B Resolving index laws for fractional exponents

    I was just thinking about this earlier and couldn't come up with a good enough resolution. I'm guessing it's a matter of convention more than anything. If we have ##x^{2} = a##, taking the principle root of both sides gives ##\sqrt{x^{2}} = \sqrt{a} \implies |x| = \sqrt{a}##. Yet evidently if...
  13. G

    Manipulating exponents: not sure how this example is worked out

    Summary: An example problem on Brilliant.org leaves out a step in solving 2^3 + 6^3 I've spend a couple days trying to understand how the second step of solving the problem 23 + 63, which is 23 + 63 = 23 + (23 * 33) = 23(1 + 33). It is not clear how this problem gets to 23(1 + 33). I can see...
  14. S

    Algebra Rational exponents in the real number system?

    Are there rigorous texts that treat the topic of raising real numbers to rational powers without treating it a special case of using complex numbers? I'm not trying to avoid the complex numbers for my own personal use! My goal is to determine whether students who have not studied complex...
  15. Edge5

    I An exponential number algebra problem

    Let a,b,c and n are real numbers.a-b = C I want to get rid of a,b and find the following expression in terms of C and n. How can I do that? (an-bn)= ? (in terms of C and n) Thank you.
  16. Monoxdifly

    MHB [ASK] Exponents and Roots Simplification problem

    The result of \frac{7x-\frac92\sqrt[6]{y^5}}{\left(x^{\frac56}-6y^{-\frac13}\right)x^{-2}} for x = 4 and y = 27 is ... a. \left(1+2\sqrt2\right)9\sqrt2 b. \left(1+2\sqrt2\right)9\sqrt3 c. \left(1+2\sqrt2\right)18\sqrt3 d. \left(1+2\sqrt2\right)27\sqrt2 e. \left(1+2\sqrt2\right)27\sqrt3 I got...
  17. F

    B How can complex numbers be elevated to complex powers?

    Hello I thought is would be fun to try a problem in which I had a complex number elevated to a complex power. To do this, I first tried to manipulate the general equation ## z^{w} ## (where ##z ## and ##w## are complex numbers) to look a bit more approachable. My work is as follows: ##z^{w}##...
  18. R

    MHB Exponents and Variable Equalities

    I am not sure how to figure this out. Basically, I know XN + XN = 2XN and that 2N + 2N = 2N+1. So, since 2 * 2 = 4 why doesn't 2N+1 = (2 * 2)N = 4N?
  19. M

    Intuitive explanation of fractional exponents?

    Homework Statement What would have caused humans to come up with fractional exponent notations? Homework EquationsThe Attempt at a Solution I understand that it makes sense to use the exponent notation when we have to multiply the same number a number of times. For example, 10^8 is the short...
  20. S

    Multiplying Algebraic Fractional Exponents

    Homework Statement a3/2a5/4 Homework EquationsThe Attempt at a Solution I'm hoping you can help. My solution to this problem would be: a3/2+5/4=a8/6=a4/3 But the answer in the back of my book is given as a11/4 I'm confused!
  21. C

    Equilibrium constant expression; why exponents?

    Hello all. I am wondering why we raise the concentration to the exponent of the stochiometric coefficient. I understand why we take the products of reactants or products. For instance, let me give an example to show where I am getting mixed up. Let's say 2A +3B = 8C as a reversible...
  22. D

    Looking for Intuitive Reasoning Behind Rational Exponents

    Solving these seem fairly simple so far. But I don't know why this works. I asked my instructor and she couldn't give me an intuitive reason as to why. Homework Statement ##\sqrt[3]y\cdot\sqrt[5]y^2## Homework Equations N/A The Attempt at a Solution $$\sqrt[3]y\cdot\sqrt[5]y^2$$ $$y^\frac...
  23. Mr Davis 97

    I Addition of exponents proof in group theory

    Just out of curiosity, what would a proof of ##a^m a^n = a^{m+n}## amount to? Of course obviously if you have n of one thing and m of another you get m+n, but I am wondering if this is rigorous enough, or if you need induction.
  24. D

    B Understanding Negative Exponents and Bases in Algebraic Expressions

    Going through a problem and and I keep getting it wrong and I'm not sure why. In a part of the problem, the expression ##\left(-3\right)\left(-r^4\right)\left(-s^5\right)## comes up and the solution that it's giving me is ##-3r^4s^5## Wouldn't the last factor be ##-s^5## since the power of a...
  25. Mr Real

    I Constant raised to complex numbers

    It's not a homework question. I just thought up a method of finding answers to problems where a number is raised to a complex number and I need to know if I am right. If we have to find e^(i), can we do it by; first squaring it to get, e^(-1) which is 1/e and then taking its square root to get...
  26. M

    MHB Rules of Exponents (4)^(1/5) * (4)^(1/5)

    Rules of Exponents (4)^(1/5) * (4)^(1/5) 4^(1/5) + (1/5) 4^(2/5) Correct? Is the answer 2^(4/5)?
  27. kolleamm

    Multiple exponents for logarithms

    Homework Statement 2^(t-1) * 6^(t-2) = 20,000 Homework EquationsThe Attempt at a Solution I have no idea how to solve this, although I do understand the basics of logarithms
  28. kolleamm

    B Chain rule for variable exponents

    I understand that when you use the chain rule you multiply the exponent by the number in front and then reduce the power by 1. So the derivative of 2x^3 = 6x^2 I'm confused now however on how you would solve something like e^-3x, the answer turns out to be -3e^-3x Am I missing a rule? Why...
  29. Cjosh

    Pulling fractional exponents out of an expression

    Homework Statement Find critical numbers of the function: F(x)=t^3/4 - 2t^1/4 Derivative I got: F'(x)=3/4 t^-1/4 - 1/2 t^-3/4 Homework EquationsThe Attempt at a Solution I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...
  30. I

    A Critical exponents - experimental values

    Hi all! For a talk I want to compare the values of the critical exponents found by Wilson and Fisher in their \epsilon = 1 paper (10.1103/PhysRevLett.28.240), with the experimental values measured up-to-date. Can anyone provide a source for these measured values (\gamma, \nu, \eta)? Thanks...
  31. T

    MHB Simplifying expression with exponents

    Supposing I have $$(-4x^n)$$ Why does it equal $(-4)^n * x^n$ and not $(-4)^n * (-x) ^ n$? When we have a negative symbol it only applies to the first item in an expression? so $-xbcw$ equals $b * c * w * (-x)$? Which means if we wanted the other items to be negative we would have to do...
  32. T

    MHB Simplifying an expression with exponents

    I have this expression $$\frac {1} { e^x + \frac {1} {e^x}}$$ and it simplifies to $$\frac {e^x} { e^{2x} + 1}$$ And I'm not sure how to get this simplification or what rules to apply to get to this simplification.
  33. L

    Multiplying different bases with different exponents

    Homework Statement Write an expression containing a single radical and simplify. Homework Equations \sqrt[4]{xy}\sqrt[3]{x^2{y}} The Attempt at a Solution I can't add the exponents and I can't multiply the bases. I can't take anything out of the radicals to make the bases the same. I have no...
  34. coolul007

    I Standard notation needed for binary numbers as exponents

    I work a lot in binary. I am organizing some of my work and need a way to write expressions. I can always create my own notation, but i would rather not invent something that already exists. 1011 is binary for 11 base 10. I use this {3,1,0} to represent the binary with just exponents. When I...
  35. A

    Lyapunov exponents of a damped, driven harmonic oscillator

    Homework Statement I am supposed to calculate Lyapunov exponent of a damped, driven harmonic oscillator given by ## \ddot{x} + 2\beta \dot{x} + \omega_0^2 x = fcos(\omega t)## Lyapunov exponent is ## \lambda ## in the equation ## \delta x(t) = \delta x_0 e^{\lambda t} ## The attempt at a...
  36. D

    B Why SQRT((-3)^2) = 3: Exploring Exponents

    I am having trouble understanding why the following expression is equal to 3: SQRT((-3)^2) What prevents me from cancelling the exponent 2 by the exponent 1/2 introduced by the square root, which yields a value of -3? Thanks for the help!
  37. DocZaius

    Tower of exponents solution approach unique to exponents?

    I saw a YouTube video presenting what seemed like a clever solution to ##x^{x^{x^{.^{.}}}} = 2## (which is to say: an infinite tower of exponents of x = 2). He said to consider just the exponents and ignore the base and realize that those exponents themselves become a restatement of the whole...
  38. Isaac0427

    Insights Complex and Irrational Exponents for the Layman - Comments

    Isaac0427 submitted a new PF Insights post Complex and Irrational Exponents for the Layman Continue reading the Original PF Insights Post.
  39. PhysicsBoyMan

    Understanding Exponent Multiplication: cnxn-1

    Homework Statement x(cnxn-1) 3. The Attempt at a Solution I know that the answer is cnxn I'm not sure why though. My thinking is that we have cnxn-1 and we multiply that by x. x times x is x2 so I'm expecting a 2 to interact with the n-1 in the exponents. I'm just not sure how n-1 interacts...
  40. C

    Examples of units with non-integer exponents

    Hi everyone, Would anyone be able to give me a few examples of places where units with non-integer exponents come up? I know of a couple right now: noise voltage density (?) in electronics has units \text{V}/\text{Hz}^\frac{1}{2} and the statcoulomb has equivalent units...
  41. M

    MHB Solve Exponent Questions: (2^2015)*(5^2019)

    How do I solve (2^2015)*(5^2019) ? Thank you, if you can teach me how to do this :) I couldn't figure out how to use LaTeX.
  42. PhysicsBoyMan

    Solving Integrals with Exponents

    Homework Statement Evaluate the integral to find the area. Homework Equations The Attempt at a Solution[/B] gifs upload So I know how to find an anti-derivative for the most part. Here it's tricky because my equation has an exponent AKA square root. I tried to use the chain rule with...
  43. D

    Greatest common factor with exponents of first input

    ok so is there a function that exists (for all intents and purposes let's call it G(x,y) )where x= a^2*b^4*c y=a^4*b^2*d G(x,y) = a^2*b^4 basically gcd, but the exponents match those of the common prime factors of the first input (x) ******** equally useful would be a function where the...
  44. K

    How Do You Solve Exponential Inequalities Using Logarithms?

    Homework Statement ##15(5^x-3^x)<16⋅15^{\frac{x}{2}}## Homework Equations Rules of logarithms The Attempt at a Solution I don't know where to start. Here's one way to start rearranging the equation: ##5^x-3^x<16⋅15^{\frac{x-2}{2}}## What's the correct way to start rearranging?
  45. T

    MHB Laws of Exponents: Understand What Your Textbook Is Saying

    In my textbook, it makes the leap from $2(2^{n + 1}) = 2^{n + 2}$ citing the laws of exponents. I'm not sure which law of exponents it is referring to. Thanks
  46. B

    Doing Fractional Exponents on Basic Scientific Calculator

    Suppose you have 8-1/3 and want a precise value for it. How would you go about calculating this on a regular scientific calculator. I punched in: 8, then the exponent button, then 1, then negative, then division, and finally 3. The calculator reads "error."
  47. P

    Dumb Question: Why Do You Add Exponents When Multiplying?

    When multiplying x2 and x5, we get x7. Quick, dumb question: Why isn't it x10 instead (since we're multiplying and not adding)?
  48. M

    Laws of Exponents: Solve Math Homework Questions

    Hello, this isn't really a physics question but I'm getting pretty desperate and you guys have always been able to help me out before :) 1. Homework Statement http://imgur.com/mYtJDLI Homework Equations I believe I'm suppose to use the Laws of Exponents. For the first set possibly but not...
  49. D

    I with multiplying rational exponents

    How can the properties of rational exponents be applied to simplify expressions with radicals and rational exponents?
  50. L

    Solve for x: 2^3+2^3+2^3+2^3=2^x

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