### Solution 1

```
u <- 4
v <- 8
u + v
```

`## [1] 12`

`u - v`

`## [1] -4`

`u * v`

`## [1] 32`

`u / v`

`## [1] 0.5`

`u^v`

`## [1] 65536`

### Solution 2

```
u <- c(4, 5, 6)
v <- c(1, 2, 3)
u + v
```

`## [1] 5 7 9`

`u - v`

`## [1] 3 3 3`

`u * v`

`## [1] 4 10 18`

`u / v`

`## [1] 4.0 2.5 2.0`

`u^v`

`## [1] 4 25 216`

### Solution 3

```
u <- c(5, 6, 7, 8)
v <- c(2, 3, 4)
u + v
```

```
## Warning in u + v: longer object length is not a multiple of shorter object
## length
```

`## [1] 7 9 11 10`

`u - v`

```
## Warning in u - v: longer object length is not a multiple of shorter object
## length
```

`## [1] 3 3 3 6`

`u * v`

```
## Warning in u * v: longer object length is not a multiple of shorter object
## length
```

`## [1] 10 18 28 16`

`u / v`

```
## Warning in u/v: longer object length is not a multiple of shorter object
## length
```

`## [1] 2.50 2.00 1.75 4.00`

`u^v`

```
## Warning in u^v: longer object length is not a multiple of shorter object
## length
```

`## [1] 25 216 2401 64`

R uses the *recycle rule* when vectors have different lengths, i.e. it re-uses elements from the shorter vector (starting at the beginning of the vector). In this case, it treats `v`

as `c(2, 3, 4, 2)`

(re-using its first element `2`

).

### Solution 4

#### Part a

```
u <- c(8, 9, 10)
v <- c(1, 2, 3)
w <- 0.5 * v
w <- u + w
w <- w^2
w
```

`## [1] 72.25 100.00 132.25`

Now check with the original approach:

```
w <- (u + 0.5 * v) ^ 2
w
```

`## [1] 72.25 100.00 132.25`

#### Part b

```
w1 <- u + 2
w2 <- u - 5
w <- w1 * w2
w <- w + v
w
```

`## [1] 31 46 63`

Now check with the original approach:

```
w <- (u + 2) * (u - 5) + v
w
```

`## [1] 31 46 63`

#### Part c

```
w1 <- u + 2
w2 <- u - 5
w2 <- w2 * v
w <- w1 / w2
w
```

`## [1] 3.333333 1.375000 0.800000`

Now check with the original approach:

```
w <- (u + 2) / ((u - 5) * v)
w
```

`## [1] 3.333333 1.375000 0.800000`

### Solution 5

#### Part a

```
w <- ((u + v) / 2) + u
w
```

`## [1] 12.5 14.5 16.5`

Now check with the original approach:

```
w <- u + v
w <- w / 2
w <- w + u
w
```

`## [1] 12.5 14.5 16.5`

#### Part b

```
w <- (u^3) / (u-v)
w
```

`## [1] 73.14286 104.14286 142.85714`

Now check with the original approach:

```
w1 <- u^3
w2 <- u - v
w <- w1 / w2
w
```

`## [1] 73.14286 104.14286 142.85714`

### Solution 6

The solution for this exercise is available in our eBook Start Here To Learn R – vol. 1: Vectors, arithmetic, and regular sequences.

### Solution 7

The solution for this exercise is available in our eBook Start Here To Learn R – vol. 1: Vectors, arithmetic, and regular sequences.

### Solution 8

The solution for this exercise is available in our eBook Start Here To Learn R – vol. 1: Vectors, arithmetic, and regular sequences.

### Solution 9

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