Exponent Definition and 220 Threads

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:





b

n


=




b
×
⋯
×
b

⏟



n



times




.


{\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. D

    MHB Sum of an Infinite Series with Real Exponent p

    $\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$ where p is any fixed real number. If this was just the telescoping series or the p-series, this wouldn't be a problem.
  2. H

    Extracting Sign, Exponent, and Mantissa from 64-bit Machine Number

    Homework Statement The problem asks us to write a MATLAB function that takes as input a vector 'v' representing a 64-bit machine number in double precision, and extract the sign (t), the exponent (c), and the mantissa (f). Homework Equations The Attempt at a Solution I'm extremely novice at...
  3. D

    Matrix Exponent Problem: Find smallest n such that Sn = I

    Homework Statement Consider the matrix cos(3*pi/17) -sin(3*pi/17) S = sin(3*pi/17) cos(3*pi/17) Does there exist a positive integer n such that Sn = I where I is the 2x2 identity? If so, what is the smallest such integer? Explain. Excuse the poor matrix...
  4. M

    Prove: Rational Number Squared Has Prime Factors w/ Even Exponent

    I have a theory that i need to prove but I am not quite sure how to mathematically prove that it is true. Theory: When you square a rational number, each of the prime factors has an even exponent. For example, 10 --> If i square 10, which is a rational number, =10^2 =(5^2 x 2^2)...
  5. mesa

    How to use the general for of the derivative formula with a fractional exponent

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  6. G

    MHB How to Use Exponents and Roots on a Scientific Calculator?

    On a scientific calculator, how do you get a root of a number for an exponiental? Ex.-.653846^1/12= I was taking 1/12 and making it so I can make it exponent for .653846. Obviously wrong. What am I doing here??
  7. C

    MHB Solving Exponents: Simplifying Radical Expressions

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  8. B

    Why does inverting the base of a negative exponent cancel the negative?

    So I understand the math, that is, how to solve these problems. But how does inverting the base cancel out the negative in the exponent? I worked out inversion to simply be a setback of 2 decimal places, e.g. 12.5% of 8 = 1, so 1/8 is .125 % of 1. But why does this nullify the negative?
  9. Z

    Fractional Exponent of Negative Numbers

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  10. M

    Abelian groups and exponent of a group

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  11. P

    How do you find out the exponent if you can't use logs?

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  12. A

    Is the Result of Complex Exponent az Multivalued?

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  13. H

    Solve an exponent equation without restricting the domain?

    It's a pretty basic equation: x = x^u, solve for u. ln x = u ln x ln x / ln x = u u = 1 However, this solution only applies for x > 0. Is there any way we could solve this equation for all values of x?
  14. J

    Negative Fractional Exponent problem

    Hi All, Can anyone walk me through this problem. This is from an old med school entry test. The answer is 81 but I can't work out how the x terms cancel. 3x(3x^ -1/3)^3 Thanks in advance John
  15. Sudharaka

    MHB Help with dividing an exponent

    MDS1005's question from Math Help Forum, Hi MDS1005, I hope you meant, that your answer is, \(\dfrac{1}{2}x^7\). If that is the case, your answer is correct and the given answer is wrong.\[\frac{5x^{2}}{10x^{-5}}=\frac{1}{2}x^{2-(-5)}=\frac{1}{2}x^7\]
  16. B

    Exponent value in Arrhenius Equation

    This may be a pretty dumb question, but how is it that as the exponent value of e decreases, the rate k increases? This seems counter-intuitive, and when I enter values for e into my calculator and compare them, the higher exponents (e.g. 50 vs. 20) give higher results.
  17. J

    CDF of a variable with a negative exponent in its PDF

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  18. C

    Calculating exponent of complex number.

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  19. N

    What is the Solution for Solving Exponential Equations with Unknown Variables?

    Solve for x, (73/10)^(x-1)+5 - 3^(x+1)=0
  20. V

    MHB Factoring when exponent is a variable

    I am trying to factor the following equation $$\large(x^{\frac{1}{n}}+a)^{n-1}$$ but the fact that the exponent is n-1 is throwing me off. How could I go about factoring out this equation? Thanks.
  21. M

    Exponent of matrix/Diagonalization of matrix with repeated eigenvalue

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  22. B

    Calculating Exponent Using Taylor Series To Given Precision

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  23. L

    Autocorrelation vs Hurst exponent. differences and similarities

    What is the relation between autocorrelation and Hurst exponent in time series analysis? which are the differences and which are the similarities? thanx
  24. P

    Approximation for -1 exponent expression

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  25. C

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  26. S

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  27. P

    Proth Primes: Coefficient & Exponent Relations

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  28. P

    Compute the limit (with variable in exponent)

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  29. M

    Calculating pressure fromV1 to V2 with a polytropic exponent

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  30. B

    Function notation minus one exponent?

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  31. J

    Easy exponent precalc problem. 4^(1+√2) * 4^(1-√2)

    Can anyone do the steps to: 4^(1+√2) * 4^(1-√2)? The book says it's 16, but I got 16^-1 steps: foil exponents: that comes out to 1-2 ? Should it not be ? then, just multiply the 4's, or do you add them?
  32. T

    Exponent Law proof Cant find anywhere?

    I have been studying the exponent laws in depth and I cannot find a formal proof on this law: exp(a+b) = exp(a)exp(b) Is it formal enough to say the following? (I think not since I give an example with numbers in it) We define: a² = a*a. For example: a^8 a^7 = a*a*a*a*a*a*a*a a*a*a*a*a*a*a =...
  33. I

    Integral of a function with exponent

    Homework Statement As part of a problem I need to evaluate, \int^{\infty}_{0}x2e-\lambdax2 Homework Equations ∫e-λx2 = 1/2 \sqrt{\pi/\lambda} ( with integration limits from 0 to ∞ ) The Attempt at a Solution I've tried integrating by parts, also tried changing variables and...
  34. G

    Finding the limit of a function with a complex exponent

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  35. E

    Expressing a complex number as an Exponent

    Homework Statement Express as z = Re[Ae^(i(\varpi t+ \alpha)] 1. z = cos(\varpi t - \pi/3) - cos (\varpit) 2. z= 2sin(\varpi t) + 3 cos (\varpi t) 3. sin(\varpi t ) - 2 cos (\varpi t - \pi/4) + cos (\varpi t) Homework Equations I used cos A + cos B; A = (a2+b2)(1/2); and tan(\theta) = y/xThe...
  36. N

    How to Evaluate lim_{x->0^{+}}(1+sinx)^{\frac{1}{\sqrt{x}}}?

    lim_{x->0^{+}}(1+sinx)^{\frac{1}{\sqrt{x}}} i get here 1^{\infty} form which states that's its some sort of exponent
  37. P

    Conjugate of amplitude with imag. # in exponent and out of exponent?

    When you have i*e^(i@) as an amplitude, when you conjugate, do both i terms switch signs? I tried this and keep getting wrong answer. Thanks in advance. BTW this has to do with spin half particles.
  38. A

    Explanation of a simple exponent rule in a derivative problem needed

    Explanation of a "simple" exponent rule in a derivative problem needed After differentiation, I get this : 9(t-2/2t+1)8 * 5/(2t+1)2 Now this simplifies into 45(t-2)/(2t+1)10 Now, I am wondering what the property is that combines the exponent from the top with the bottom to get an power of...
  39. N

    How can I simplify and solve this expression with negative exponents?

    Homework Statement http://i51.tinypic.com/2zyfyaa.jpg Question 3 l ) Homework Equations The Attempt at a Solution (6a^-2b^-3)^2 (__________) 1/6a^2 * 1/b^3 / 1/2a^2 * 1/b^1 (2a^2b^-1 ) I multiplied the top half together to give me 1/ 36a^4b^2...
  40. N

    Exponent Laws Practice: Simplifying Fractional Exponents

    Homework Statement 2^-4 + 2^-6 _____________ 2^-3 Homework Equations The Attempt at a Solution Ive tried several times, I do the reciprical of all numbers ie) 1/2^4 + 1/2 ^6 ___________________ 2^3 Which gives me The wrong answer, cause i do the addiition first...
  41. N

    Solving Exponent Laws: 2^3 / 2^0 - 2^-1 Question on Homework

    Homework Statement 2^3 / 2^0 -2^-1 Homework Equations solving using exponent laws The Attempt at a Solution i move the negetive one over and make it 2^3 /2^0 = 1/2^1 I get 8 on the left side, and 2 on the right, bring the 6 from the right to the left and get 2. The answer is...
  42. M

    Getting stress-strain data using the 'strain hardening exponent'.

    Hi guys, I am trying to generate tabular stress-strain data from the data given in this table for a grade of aluminium, The elastic region is simple but I don't understand how to use the 'strain hardening exponent' to generate stress-strain data for the plastic region. From what I...
  43. D

    Lyapunov exponent - order of magnitude

    Hello, Analyzing data from a chaotic pendulum, I calculated the Lyapunov exponent to be somewhere around 10^5 . While my gut tells me something is wrong with this number , i failed to find any information regarding the order of magnitude of Lyapunov exponents and their meaning. Can someone...
  44. P

    Why does (e^{i\alpha})^2 always equal 1?

    In solutions to a problem I was working on, I saw that when an expression such as e^{i\alpha} , alpha being an angle (in polar coordinates), was squared, the expression goes to unity, ie (e^{i\alpha})^2=1 But I see no reason to think that \alpha is a multiple of \pi. Could there be any other...
  45. B

    Getting Rid of Negative Exponent

    Getting Rid of Negative Exponent! Homework Statement So I am solving a Diff Eq ( separable equation) and I am down the the last step. After integrating both sives and solving for y I am left with y^(-2) on the left hand side. How do I finalize it and get rid of negative 2 expononent...
  46. M

    Difference between unit-fractional exponent and root sign

    We were having our maths class a weeks ago and while studying quadratic equations our teacher asked us if there was an difference between x^{1/2} and \sqrt{x}. Up until then I had always assumed them to be the same thing. However he said that there was a difference and said that he would tell...
  47. madmike159

    C/C++ C++: Extracting Exponent Value from a Double

    Is there a way to look at a double value in c++ and read the exponent only and put this into another variable? So if I had 3.14159x10^300 I could take 300, put it into an int Then divide 3.14159x10^300 by 1x10^(the int value)
  48. Drakkith

    How do you square a negative exponent for kinetic energy calculation?

    Trying to find out the kinetic energy that a 1kg ball gives the Earth if dropped from 100 ft. I just don't know how to square a negative exponent. 4x10^-24 squared is? I was thinking it would be 4x10^-48, but I really don't know. If that is correct, would the result be different if the exponent...
  49. G

    Proving Exponent of Finite Groups Divides Order

    Homework Statement prove that every finite group has exponent that divides the order of the group Homework Equations The Attempt at a Solution Given G is a finite group and x \in G. Suppose x has order m, then <x> = {e,x,x^2...x^(m-1)} and so \left|<x>\right| = m so by...
  50. K

    I have a variable as an exponent, how do I isolate it from everything else?

    Homework Statement .75 = 1 - (1 - p)^n I want to get the n alone, and find n in terms of p. Homework Equations The Attempt at a Solution I simplified it to this: 1 - p = (.25)^(1/n) Basically, I just moved numbers around. I do not know how to get the n by itself without...
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