Evaluate expressions involving squares and cubes of integers
Squares and cubes
Squaring and cubing numbers builds on a number of concepts that you have learnt already, including how to add and subtract integers, how to multiply and divide integers, as well as our order of operations.
What does squaring mean?
Squaring a number means multiplying it by itself.
Examples
This diagram shows 12, 22, 32,, 42 and 52
$3^2=3\times3$32=3×3 $=$= $9$9
$7^2=7\times7$72=7×7 $=$= $49$49
$11^2=11\times11$112=11×11 $=$= $121$121
What about negative squares?
We learnt in More Multiplication that when we multiply two negative numbers the product is positive. This means that if we square a negative number we end up with a positive result.
Examples
$\left(-3\right)^2=\left(-3\right)\times\left(-3\right)$(−3)2=(−3)×(−3) $=$= $9$9
$\left(-9\right)^2=\left(-9\right)\times\left(-9\right)$(−9)2=(−9)×(−9) $=$= $81$81
$-3^2$−32 is not the same as $\left(-3\right)^2$(−3)2
$-3^2$−32 means $-\left(3^2\right)$−(32) or $-1\times\left(3\times3\right)$−1×(3×3), which gives us an answer of $-9$−9 because we are taking the square of $3$3 and then multiplying by $-1$−1.
$\left(-3\right)^2$(−3)2 means $\left(-3\right)\times\left(-3\right)$(−3)×(−3), which gives us an answer of $9$9 because the brackets mean we are taking the square of $-3$−3.
What does cubing mean?
Cubing a number means multiplying it by itself twice.
Examples
$2^3=2\times2\times2$23=2×2×2 $=$= $8$8
$5^3=5\times5\times5$53=5×5×5 $=$= $125$125
$10^3=10\times10\times10$103=10×10×10 $=$= $1000$1000
But what happens when you cube a negative number?
When we cube a negative number, we'll end up with a negative number. How does that work? Let's take a look.
Example
Question 1
Evaluate: $\left(-2\right)^3$(−2)3
Think: $\left(-2\right)^3=-2\times\left(-2\right)\times\left(-2\right)$(−2)3=−2×(−2)×(−2). We know multiplying two negative numbers will give us a positive answer, so $-2\times\left(-2\right)=4$−2×(−2)=4.
So, when we multiply the first two negative numbers, we get a positive answer.
Then we multiply a positive and a negative number, which gives us a negative answer.
Do:
| $\left(-2\right)^3$(−2)3 | $=$= | $-2\times\left(-2\right)\times\left(-2\right)$−2×(−2)×(−2) |
| $=$= | $4\times\left(-2\right)$4×(−2) | |
| $=$= | $-8$−8 |
Question 2
Evaluate $-3^3$−33.
Questions using order of operations
Now we need to consider our order of operations again. Where do squares and cubes (and other powers) fit in?
- Perform all operations within grouping symbols.
- Evaluate all squares and cubes (and other powers).
- Do all multiplications and divisions in the order in which they occur, working from left to right.
- Do all additions and subtractions in the order in which they occur, working from left to right.
Let's look at some examples that put all these rules together.
Examples
Question 3
Evaluate $\left(-12\right)^2-\left(-5\right)^2$(−12)2−(−5)2.
Question 4
Evaluate: $-3^3+2\times5^2-\left(-4\right)^2$−33+2×52−(−4)2
Think: Following the order of operations, we want to work out the squares first. Note that we don't want to evaluate $2\times5$2×5 as the squares take priority. We can then perform the multiplication, and lastly add and subtract working from left to right.
Do:
| $-3^3+2\times4^2-\left(-5\right)^2$−33+2×42−(−5)2 | $=$= | $-27+2\times16-25$−27+2×16−25 |
| $=$= | $-27+32-25$−27+32−25 | |
| $=$= | $-20$−20 |
question 5
Evaluate: $12^2-\left(-2\right)^3+27$122−(−2)3+27
Think: Work out the cube of $\left(-2\right)$(−2) first, and then perform the subtraction and addition working from left to right.
Do:
| $12^2-\left(-2\right)^3+27$122−(−2)3+27 | $=$= | $144-\left(-8\right)+27$144−(−8)+27 |
| $=$= | $144+8+27$144+8+27 | |
| $=$= | $179$179 |