1: No difference. It is kept around to allow old S-code to continue to function. This is documented a "Note" in ?Math?Arithmetic

2: Yes: But you already know it:

`^`(x,y)
#[1] 1024

In R the mathematical operators are really functions that the parser takes care of rearranging arguments and function names for you to simulate ordinary mathematical infix notation. Also documented at ?Math.

Edit: Let me add that knowing how R handles infix operators (i.e. two argument functions) is very important in understanding the use of the foundational infix "[[" and "["-functions as (functional) second arguments to lapply and sapply:

> sapply( list( list(1,2,3), list(4,3,6) ), "[[", 1)
[1] 1 4
> firsts <- function(lis) sapply(lis, "[[", 1)
> firsts( list( list(1,2,3), list(4,3,6) ) )
[1] 1 4
Answer from IRTFM on Stack Overflow
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CRAN
cran.r-project.org › web › packages › pwrss › vignettes › examples.html
Practical Power Analysis in R
September 16, 2025 - These generic functions compute and return statistical power with the option to generate Type I and Type II error plots when test statistics and degrees of freedom are available. Users can input test statistics manually, valuable for custom designs outside the {pwrss} package’s scope, or extract them from statistical software output.
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GeeksforGeeks
geeksforgeeks.org › r language › logarithmic-and-power-functions-in-r-programming
Logarithmic and Power Functions in R Programming - GeeksforGeeks
June 1, 2020 - First is the Logarithm, to which the general way to calculate the logarithm of the value in the base is with the log() function which takes two arguments as value and base, by default it computes the natural logarithm and there are shortcuts for common and binary logarithm i.e. base 10 and 2. Value can be number or vector. Second is the Power, to calculate a base number raised to the power of exponent number.
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ProjectPro
projectpro.io › recipes › exponential-operation-r
How to compute exponents in R - Projectpro
January 5, 2024 - In R, the exponentiation operation calculates the power of one numeric variable raised to the power of another.
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Cyclismo
cyclismo.org › tutorial › R › power.html
11. Calculating The Power Of A Test — R Tutorial - Cyclismo.org
The power is the probability that we do not make a type II error so we then take one minus the result to get the power. We can fail to reject the null hypothesis if the sample happens to be within the confidence interval we find when we assume that the null hypothesis is true.
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Statsthinking21
statsthinking21.github.io › statsthinking21-R-site › statistical-power-in-r.html
Chapter 9 Statistical power in R | An R companion to Statistical Thinking for the 21st Century
In this case, let’s say that we wish to perform a two-sample t-test. # create a function get the power value and # return as a tibble get_power <- function(df){ power_result <- pwr.t.test(n=df$sample_sizes, d=df$effect_sizes, type='two.sample') df$power=power_result$power return(df) } # run get_power for each combination of effect size # and sample size power_curves <- input_df %>% do(get_power(.)) %>% mutate(effect_sizes = as.factor(effect_sizes))
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Ladal
ladal.edu.au › pwr.html
Power Analysis in R – LADAL
Power analysis is a method primarily used to determine the appropriate sample size for empirical studies. This tutorial is aimed at intermediate and advanced users of R with the aim of showcasing how to perform power analyses for basic inferential tests using the pwr package (Champely 2020) ...
Find elsewhere
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R-bloggers
r-bloggers.com › r bloggers › power analysis in statistics with r
Power analysis in Statistics with R | R-bloggers
May 8, 2021 - Hence α may be relatively larger than β. ... For testing a hypothesis H0 against H1, the test with probabilities α and β of Type I and Type II errors respectively, the quantity (1- β) is called the power of the test.
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SQLPad
sqlpad.io › tutorial › exponents-r-comprehensive-guide
Exponents in R: A Comprehensive Guide
May 2, 2024 - This operator is the cornerstone for performing basic to advanced exponential calculations. Let's dive into its usage with practical examples to illustrate its versatility. Basic Exponentiation: To start, raising a number to a power is as simple as 2^3, which will return 8.
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RDocumentation
rdocumentation.org › packages › extras › versions › 0.8.0 › topics › pow
pow function - RDocumentation
R equivalent to the power function. pow(x, n) A numeric atomic object of x raised to n. x · A numeric atomic object of the base. n · A numeric atomic object of the exponent. Wrapper on x^n. Other translations: exp10(), exp2(), fabs(), ilog(), ilog10(), ilog2(), ilogit(), inv_logit(), invlogit(), ...
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ProgrammingR
programmingr.com › home
Exponents in R [How Raise A Number to a Power in R Code] - ProgrammingR
April 29, 2022 - There are two ways of doing exponents in r. The first has the format of x^y where “x” is the number that is going to be raised to the “y” power. This version is the most common way in programming of doing exponents. The second has the format of x**y where “x” is the number that ...
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Egap
egap.org › resource › script-power-analysis-simulations-in-r
Script: Power Analysis Simulations in R – EGAP
The graph below shows the output of this simulation — at any sample size, the covariate-adjusted model does better than the unadjusted model. In fact, the unadjusted model requires three times as many subjects to achieve 80% power than does the covariate-adjusted model.
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Resbaz
resbaz.github.io › 2014-r-materials › lessons › 01-intro_r › r-basics.html
Getting started in R
The [1] preceding this we will explain in a minute. For now, think of it as something that indicates output. Order of operations is same as in maths class (from highest to lowest precedence) ... The "caret" symbol (or "hat") is the exponent (to-the-power-of) operator (read x ^ y as "x to the ...
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R-statistics
r-statistics.co › home › power analysis in r: calculate the sample size you need before you collect any data
Power Analysis in R: Calculate the Sample Size You Need Before You Collect Any Data
6 days ago - The pwr package in R solves for any one of four quantities (sample size, effect size, significance level, power) given the other three, across t-tests, ANOVA, correlation, and regression.
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GeeksforGeeks
geeksforgeeks.org › data analysis › power-analysis-in-statistics-with-r
Power Analysis in Statistics with R - GeeksforGeeks
July 23, 2025 - First installed the pwr package, do so with the following command: ... # Parameters for two-sample t-test effect_size_t <- 0.5 # Moderate effect size (Cohen's d) alpha_t <- 0.05 # Significance level power_t <- 0.8 # Desired power # Calculate required sample size sample_size_t <- pwr.t.test(d = effect_size_t, sig.level = alpha_t, power = power_t, type = "two.sample")$n # Output the result cat("Sample Size for Two-Sample t-Test:", sample_size_t, "\n")
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Reddit
reddit.com › r/rstats › power calculations
r/rstats on Reddit: Power calculations
November 13, 2023 -

So, I need to calculate power, or rather, how many samples we need to show a particular size of effect. Unfortunately, this is one of the areas that was left out of my education.

Could you please dear rstats people point me on a good source for this?

Or should I just do a simulation given a particular model and assumption so that I can calculate the required number of samples for different scenarios? (anything from anova, or t-test if you want, populations with different variance, interactions, methods like CART, RF, boosting...)?

So far I feel like one equation from Wikipedia and table with different effect sizes and required populations, and then invest time into simulations might be more sustainable deal.

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Reddit
reddit.com › r/askmath › what does power on r mean?
r/askmath on Reddit: What does power on R mean?
August 2, 2022 -

For context, I am not mathematician, I'm studying engineering. I'm often seeing some power over R (set of real numbers). Every text and note I've found so far just casually starts using it. I do not understand it's meaning. I can infer a little from context but can't confirm since Google search isn't helping. I'd be very thankful if someone could explain it for me.

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RDocumentation
rdocumentation.org › packages › stats › versions › 3.6.2 › topics › power
power function - RDocumentation
To calculate the power of a test, see various functions in the stats package, e.g., power.t.test.
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ETH Zurich
stat.ethz.ch › R-manual › R-devel › library › stats › html › power.t.test.html
R: Power calculations for one and two sample t tests
Compute the power of the one- or two- sample t test, or determine parameters to obtain a target power.