Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- What is 2³?
Answer
8: 2 × 2 × 2. - Work out (−5) × (−5).
Answer
25: the same signs give a positive answer. - Work out 3 + 4 × 2.
Answer
11: multiply before you add.
Words used in this lesson
- power (exponent)
- how many times the base is written as a factor. For example: in 2⁵ the base 2 is a factor 5 times: 2 × 2 × 2 × 2 × 2 = 32.
- base
- the number being multiplied. For example: in (−2)⁵ the base is −2.
- even / odd
- divides exactly by 2 / does not. For example: 4 is even, 5 is odd.
- brackets
- show what belongs together and is done first. For example: (−3)².
- order of operations
- the agreed order for calculating. For example: brackets, powers, × and ÷, + and −.
What does the power touch?
A power is repeated multiplication: 3² = 3 × 3 = 9. With a negative number, everything depends on what the power is attached to.
- (−3)² means (−3) × (−3) = 9. The brackets say the whole of −3 is squared.
- −3² means −(3²) = −(3 × 3) = −9. The power touches only the 3; the minus sign waits until afterwards.
Powers come before negation for the same reason they come before subtraction: the order of operations. In the figure, the top bar in each row has the brackets and the bottom one does not. Before you move anything, guess which rows will have the two bars pointing the same way.
The power 4 is even, so (−2)⁴ = 16 is positive, but −2⁴ means −(2⁴) = −16.
Show me the steps
- k = 2, with brackets: (−2)¹ = −2, (−2)² = 4, (−2)³ = −8, (−2)⁴ = 16, (−2)⁵ = −32. The sign flips at every step.
- k = 2, without brackets: −2¹ = −2, −2² = −4, −2³ = −8, −2⁴ = −16, −2⁵ = −32. Always negative.
- They agree for the odd powers 1, 3 and 5, and are opposites for the even powers 2 and 4.
- k = 3: (−3)² = 9 but −3² = −9; (−3)⁴ = 81 but −3⁴ = −81.
A negative number raised to an even power is positive; to an odd power it is negative. Negative factors cancel in pairs. With an even number of them nothing is left over; with an odd number one is left, so the answer is negative.
Without brackets the power belongs to the number alone: −kⁿ is always negative, whatever the power.
The order of operations, with integers
The order is the one you already know: brackets, then powers, then multiplication and division (left to right), then addition and subtraction (left to right). The signs simply come along with each number.
Your go. The first step is given. Work out 10 − (−4)².
- Power first: (−4)² = 16.
- Finish it, then check.
Check the answer
10 − 16 = −6.
The most common slip: reading −5² as (−5)². Ask yourself what the power is touching before you calculate. If it touches only the 5, square the 5 and then put the minus sign back.
With a calculator
On most scientific calculators, press − 3 x² = and the screen shows −9: the calculator squares first and applies the minus sign afterwards, exactly as the convention says. Press ( − 3 ) x² = and it shows 9.
So the calculator agrees with the rule rather than replacing it. Use it to check a power you have already worked out, and type the brackets whenever you mean the negative number itself. Some calculators have a separate (−) key for a negative number: it still needs the brackets.
Check before you go on
1. What is −4²?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Work out (−4)².
3. Work out (−1)⁷.
Level 2 · Building
4. Work out (−2)⁵.
5. Work out −2 + 5 × (−3).
6. Work out (−3)² − 4 × (−2).
Level 3 · Stretch
7. Work out (−1)¹⁰⁰.
8. How much greater is (−3)⁴ than −3⁴?
Something to think about
Can two different integers have the same square? Can two different integers have the same cube?
One way to think about it
Squares, yes: 3² = 9 and (−3)² = 9. Cubes, no: a cube keeps the sign of the number, so 3³ = 27 but (−3)³ = −27, and different integers always have different cubes.
Exit ticket
9. Sam types −5² into a scientific calculator and it shows −25. Sam says the calculator is wrong. Who is right?