Directed Numbers

Lesson

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.

*Squaring* a number means *multiplying it by itself.*

$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

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.

$\left(-3\right)^2=\left(-3\right)\times\left(-3\right)$(−3)2=(−3)×(−3) $=$= $9$9

$\left(-9\right)^2=\left(-9\right)\times\left(-9\right)$(−9)2=(−9)×(−9) $=$= $81$81

But be careful!

$-3^2$−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.

*Cubing* a number means *multiplying it by itself twice*.

$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

When we cube a negative number, we'll end up with a negative number. How does that work? Let's take a look.

**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 |

**Evaluate **$-3^3$−33.

Now we need to consider our order of operations again. Where do squares and cubes (and other powers) fit in?

Order of operations!

- 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.

**Evaluate **$\left(-12\right)^2-\left(-5\right)^2$(−12)2−(−5)2.

**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 |

**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 |

Understand operations on fractions, decimals, percentages, and integers