Multiplying Integers
Multiplying Integers in NSW Year 8 maths explains why a negative times a negative is positive, then turns it into a quick rule (NESA outcome MA4‑INT‑C‑01): multiply the sizes and count the negative factors. An even count gives a positive answer and an odd count gives a negative one.
You will multiply strings of three or more integers, choose an efficient order, and evaluate powers such as \((-3)^4\) and \(-3^4\), which are not the same. You will find a missing factor or index and spot the pattern in powers of \(-1\). A weekly phone plan charge, a dam level falling each day and salty water cooling in a freezer are all products of integers. It is part of Year 8 Working with Numbers.
Every question with a fully worked solution.
- Multiplying Integers - Video - Multiplying Integers | How to Multiply Positive and Negative Integers Watch
Theory
Multiply the sizes, then decide the sign: the same signs give a positive answer and different signs give a negative one. With many factors, count the negatives: an even count gives a positive, an odd count a negative. This Year 8 maths page also covers powers of negative numbers.
Sign rules. \(4 \times 3 = 12\), \(4 \times (-3) = -12\), \(-4 \times 3 = -12\) and \(-4 \times (-3) = 12\). Same signs: positive. Different signs: negative.
Why a negative times a negative is positive. \(4 \times (-7)\) is four lots of \(-7\): \(-7 - 7 - 7 - 7 = -28\). Multiplying by \(-4\) gives the opposite of multiplying by \(4\), so \(-4 \times (-7) = 28\).
Several factors. Count how many are negative. Even (\(0, 2, 4, \ldots\)) gives a positive product; odd (\(1, 3, 5, \ldots\)) gives a negative one. Then multiply the sizes.
Any order. A product may be rearranged, so pair factors that make a round number: \(4 \times 25 = 100\), \(8 \times 125 = 1000\). Bigger sizes: split into tens and ones, \(23 \times 6 = 120 + 18 = 138\), and decide the sign last.
Powers. The index counts the factors: \((-2)^3 = (-2) \times (-2) \times (-2) = -8\). An even index gives a positive, an odd index a negative. Without brackets only the number is raised: \(-7^2 = -49\), but \((-7)^2 = 49\).
Missing factor. Multiply the known factors, find the size by dividing, then choose the sign that fits.
Sign rules for multiplying:
| Signs | Example | Answer |
|---|---|---|
| positive × positive | \(7 \times 3\) | \(21\) |
| positive × negative | \(7 \times (-3)\) | \(-21\) |
| negative × positive | \(-7 \times 3\) | \(-21\) |
| negative × negative | \(-7 \times (-3)\) | \(21\) |
How to multiply integers
- Count the negative factors (a power counts as many factors as its index).
- Decide the sign: even count positive, odd count negative.
- Multiply the sizes, in any convenient order.
How to find a missing factor
- Multiply the known factors.
- Divide the sizes to find the size of the missing factor.
- Choose the sign that gives the right answer, then check.
(i) \(7 \times (-3)\)
(ii) \(-7 \times 3\)
(iii) \(-7 \times (-3)\)
| \(7 \times (-3)\) | \(=\) | \(-21\) |
| \(-7 \times 3\) | \(=\) | \(-21\) |
| \(-7 \times (-3)\) | \(=\) | \(21\) |
(i) \(-21\)
(ii) \(-21\)
(iii) \(21\) (same signs)
(i) \(-23 \times 6\)
(ii) \(-14 \times (-30)\)
| \(23 \times 6\) | \(=\) | \(20 \times 6 + 3 \times 6 = 138\) |
| \(14 \times 30\) | \(=\) | \(420\) |
(i) Different signs: \(-138\)
(ii) Same signs: \(420\)
(i) \((-7)^2\) and \(-7^2\)
(ii) \(-(-3)^3\)
| \((-7)^2\) | \(=\) | \((-7) \times (-7) = 49\) |
| \(-7^2\) | \(=\) | \(-(7 \times 7) = -49\) |
| \((-3)^3\) | \(=\) | \(-27\) |
| \(-(-3)^3\) | \(=\) | \(-(-27) = 27\) |
(i) \(49\) and \(-49\)
(ii) \(27\)
Two negative factors: even, so the answer is positive. Pair \(5 \times 2 = 10\) and \(9 \times 4 = 36\).
| \(10 \times 36\) | \(=\) | \(360\) |
\(360\)
(i) What is the change over the next \(6\) hours?
(ii) \(4\) hours ago is a time of \(-4\). How much water was in the tank then?
| \(6 \times (-45)\) | \(=\) | \(-270\) |
| \(-4 \times (-45)\) | \(=\) | \(180\) |
| \(900 + 180\) | \(=\) | \(1080\) |
(i) \(-270\) L
(ii) \(1080\) L. The tank was fuller earlier.
(i) \({?} \times (-6) \times (-2) = -84\)
(ii) \((-4)^{?} = 256\)
| \(-6 \times (-2)\) | \(=\) | \(12\) |
| \(84 \div 12\) | \(=\) | \(7\) |
(i) \(-7\). Check: \(-7 \times 12 = -84\).
(ii) \(256\) is positive, so the index is even: \((-4)^4 = 16 \times 16 = 256\), so \({?} = 4\).
Common pitfalls
Frequently asked questions
Why is a negative times a negative positive?
Multiplying by \(-1\) gives the opposite. \(4 \times (-7) = -28\), so \(-4 \times (-7)\) is the opposite of \(-28\), which is \(28\).
How do you work out the sign of a long product?
Count the negative factors. Even gives positive; odd gives negative.
Is a negative number to the power 4 positive or negative?
Always positive, because \(4\) is even: \((-4)^4 = 256\).
What is negative one to an even power?
\(1\). An odd power of \(-1\) is \(-1\).
What is the difference between -5 squared and (-5) squared?
\(-5^2 = -25\) but \((-5)^2 = 25\).