Problem 1

First:

x <- 1.1
a <- 2.2
b <- 3.3

For a:

z <- x^(a^b)
print(z)

## [1] 3.61714

For b:

z <- ((x^a)^b)
print(z)

## [1] 1.997611

For c:

z <-3*(x^3)+2*(x^2)+1
print(z)

## [1] 7.413

Problem 2

For a:

vector_a <- c(1:8, 7:1)
print(vector_a)

##  [1] 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

For b:

vector_b1 <- 1:5
vector_b2 <- rep(x=vector_b1,times=vector_b1)
print(vector_b2)

##  [1] 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5

OR:

vector_b2 <- rep(x=1:5,times=1:5)
print(vector_b2)

##  [1] 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5

For c:

vector_c3 <- rep(x=5:1,times=1:5)
print(vector_c3)

##  [1] 5 4 4 3 3 3 2 2 2 2 1 1 1 1 1

vectorr <- runif(2)
names(vectorr) <- c("x","y")
print(vectorr)

##          x          y 
## 0.41905218 0.03187638

r <- ((vectorr[["x"]]^2) + (vectorr[["y"]]^2))^(1/2)
print(r)

## [1] 0.4202628

theta <- atan(vectorr[["y"]]/vectorr[["x"]])
print(theta)

## [1] 0.07592159

queue <- c("sheep", "fox", "owl", "ant")
queue <- c(queue,"snake")  
print(queue)

## [1] "sheep" "fox"   "owl"   "ant"   "snake"

queue <- queue[queue!="sheep"]
print(queue)

## [1] "fox"   "owl"   "ant"   "snake"

queue <- c("donkey", queue)
print(queue)

## [1] "donkey" "fox"    "owl"    "ant"    "snake"

queue <- queue[queue!="snake"]
print(queue)

## [1] "donkey" "fox"    "owl"    "ant"

queue <- queue[queue!="owl"]
print(queue)

## [1] "donkey" "fox"    "ant"

queue <- c(queue[queue!="ant"],"aphid","ant")
print(queue)

## [1] "donkey" "fox"    "aphid"  "ant"

I do recognize that I can only do it this exact way because I know the ant is already last

which(queue=="aphid")

## [1] 3

vec_1 <- 1:100
vec_2 <- vec_1[(vec_1%%2)!=0&(vec_1%%3)!=0&(vec_1%%7)!=0]
print(vec_2)

##  [1]  1  5 11 13 17 19 23 25 29 31 37 41 43 47 53 55 59 61 65 67 71 73 79 83 85
## [26] 89 95 97