Surds

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Simplify negative square and cube roots

Lesson

Finding the square root of a number is the inverse (or opposite) operation of squaring a number.

Remember squaring a number means multiplying a number by itself (e.g. $3^2=3\times3$32=3×3). The square root of a number is the number that you need to multiply by itself to get the original number. What? Let's look at an example:

**Evaluate:** $\sqrt{25}$√25

**Think:** $5^2=25$52=25

**Do:** $\sqrt{25}=5$√25=5

So what happens when we bring in negative numbers?

Firstly, let me point out at this stage, you will NOT have to find the square root of a negative number. Why?

Remember when we were squaring numbers- multiplying two negative numbers always gives a positive answer. So we never get a negative answer. The answer to a negative square root is called an *imaginary number*.

Let's look at questions where we can include a negative symbol.

**Evaluate:** $-\sqrt{64}$−√64

**Think:** $8^2=64$82=64

**Do:** $-\sqrt{64}=-8$−√64=−8

We can find the cubed root of a negative number (because when we cube a negative number, we get a negative answer).

**Evaluate:** $\sqrt[3]{-8}$^{3}√−8

**Think: **$\left(-2\right)^3=-8$(−2)3=−8

**Do:** $\sqrt[3]{-8}=-2$^{3}√−8=−2

Here's another one:

**Evaluate:** $-\sqrt[3]{-216}$−^{3}√−216

**Think:** $6^3=216$63=216

**Do:**

$-\sqrt[3]{-216}$−^{3}√−216 |
$=$= | $-1\times\left(-6\right)$−1×(−6) |

$=$= | $6$6 |

Building on the concepts that you have learnt already, including how to add and subtract integers, how to multiply and divide integers, as well as order of operations, you can do more complex questions using negative square and cubed roots.

**Evaluate: **$\sqrt[3]{-64}\times\sqrt{64}$^{3}√−64×√64

**Think:** The cube root of $-64$−64 is $-4$−4 and the square root of $64$64 is $8$8.

**Do:**

$\sqrt[3]{-64}\times\sqrt{64}$^{3}√−64×√64 |
$=$= | $-4\times8$−4×8 |

$=$= | $-32$−32 |

Here's another one

**Evaluate:** $\left(-\sqrt{85+15}\right)\times\left(-\sqrt[3]{-125}\right)$(−√85+15)×(−^{3}√−125)

**Think: **The square root of $100$100 is $10$10. The cubed root of $-125$−125 is $-5$−5.

**Do:**

$\left(-\sqrt{85+15}\right)\times\left(-\sqrt[3]{-125}\right)$(−√85+15)×(−^{3}√−125) |
$=$= | $\left(-\sqrt{100}\right)\times\left(-1\right)\times\left(-5\right)$(−√100)×(−1)×(−5) |

$=$= | $-10\times5$−10×5 | |

$=$= | $-50$−50 |

Evaluate $\sqrt[3]{-64}$^{3}√−64

Evaluate $\sqrt[3]{-125}\times\sqrt[3]{27}$^{3}√−125×^{3}√27

Manipulate complex numbers and present them graphically

Apply the algebra of complex numbers in solving problems