Two roots, one plus one minus
8 × 8 = 64, and so is −8 × −8.
The other 8
We know 8 × 8 = 64. Now think about −8 × −8.
What is −8 × −8?
What this lesson covers
The idea
Every perfect square has two integer square roots, one positive and one negative, written √n² = ±n; often only the positive square root, denoted by √, is considered.
The other 8
We know 8 × 8 = 64. Now think about −8 × −8.
What is −8 × −8?
- −64
- 64
- 0
Find both numbers
Drag the token along the number line. Find both numbers whose square is the target.
Plus and minus
Every perfect square has two integer square roots, one positive and one negative: √n² = ±n. Often only the positive square root, written with √, is used.
In this chapter we use only the positive square root, so √64 = 8.
- Square | Square roots
- 25 | +5 and −5
- 64 | +8 and −8
- 100 | +10 and −10
Notes
Every perfect square has two integer square roots, one positive and one negative: √n² = ±n. Usually only the positive root, written √, is taken.
Check yourself
Which numbers have a square equal to 81?
What is (−7)²?
Answer: 49
−7 × −7 = 49, the same as 7 × 7.
What is the negative square root of 169?
Answer: -13
(−13) × (−13) = 169, so the roots of 169 are +13 and −13.
- 9 only. (−9) × (−9) = 81 too. A negative times a negative is positive.
- −9 only. 9 × 9 = 81 too.
- 9 and −9 — correct. Yes! 9² = 81 and (−9)² = 81, so 81 has the two roots +9 and −9.
- 81. 81 × 81 is much bigger than 81.