Question

Simplify the following.

(i) (−1) × 2 × (−3)

(ii) 1 × (−2) × 3

(iii) (−2) × (−3) × (−4)

(iv) (−1) × (−1)

(v) (−1)³

(vi) (−1)⁵

(vii) (−1)⁹⁹

Answer 1

(i)

(−1) × 2 × (−3) = (−1) × (2 × (−3))

= (−1) × (−6) {x × (−y) = −(xy), for all positive numbers x and y}

= 1 × 6 {(−x) × (−y) = xy, for all positive numbers x and y}

= 6

(ii)

1 × (−2) × 3 = (1 × (−2)) × 3 {x × (−y) = −(xy), for all positive numbers x and y}

= (−2) × 3

= −(2 × 3) {(−x) × y = −(xy), for all positive numbers x and y}

= −6

(iii)

(−2) × (−3) × (−4) = ((−2) × (−3)) × (−4)

= 6 × (−4) {(−x) × (−y) = xy, for all positive numbers x and y}

= −(6 × 4) {x × (−y) = −(xy), for all positive numbers x and y}

= −24

(iv)

(−1) × (−1) = 1 × 1 {(−x) × (−y) = xy, for all positive numbers x and y}

= 1

(v)

(−1)³ = (−1) × (−1) × (−1)

= (1 × 1) × (−1) {(−x) × (−y) = xy, for all positive numbers x and y}

= 1 × (−1)

= −(1 × 1) {x × (−y) = −(xy), for all positive numbers x and y}

= −1

(vi)

(−1)⁵ = (−1) × (−1) × (−1) × (−1) × (−1)

= (1 × 1) × (−1) × (−1) × (−1) {(−x) × (−y) = xy, for all positive numbers x and y}

= 1 × (−1) × (−1) × (−1)

= −(1 × 1) × (−1) × (−1) {x × (−y) = −(xy), for all positive numbers x and y}

= (−1) × (−1) × (−1)

= (1 × 1) × (−1) {(−x) × (−y) = xy, for all positive numbers x and y}

= 1 × (−1)

= −(1 × 1) {x × (−y) = −(xy), for all positive numbers x and y}

= −1

(vii)

(−1)⁹⁹ = (−1) × (−1) × (−1) × (−1) × (−1)….. 99 times

We have calculated that (−1)³ = −1

(−1)⁵ = −1

Thus, when (−1) is raised to any odd number, the result is −1.

As 99 is an odd number, (−1)⁹⁹ = −1