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# Indices: Negative Numbers

### Indices: Negative Numbers

We can have negative numbers as the base number and must be able to calculate indices with these.

An index number is a base number raised to a power. How do you write $6$ squared?

We can also raise negative numbers to a power. What is the correct way of writing minus $3$ squared?

Brackets are needed to be clear that we are squaring the negative number.

Without brackets, we square the number first and then make negative.

What is another way of writing $(-3)^2$?

What is $(-3)^2$?

Work out $(-6)^2$

There is a pattern when squaring negative numbers. What type of number is the answer?

The square of a negative number is *always* positive.

For example $(-4)^2 = (-4) \times (-4) = 16$.

However, this isn't the case when **cubing** negative numbers.

Let's find $(-3)^3$

Cubing multiplies a number by itself three times

So $(-3)^3$ is the same as $(-3) \times (-3) \times (-3)$

What is $(-3) \times (-3)$?

What is $9 \times (-3)$?

Cubing negative numbers always returns a negative

This is true for every negative number. Nice 👍

What is $(-5)^3$?

We have seen that

Squaring a negative number gives a **positive** answer
Cubing a negative number gives a **negative** answer

What about other powers? What is $(-3)^4$?

The answer to $(-3)^4$ was positive. Will the answer to $(-3)^5$ be positive or negative?

We have a pattern here.

Negative base number + **EVEN** power = **Positive number**
Negative base number + **ODD** power = **Negative number**

Is $(-143)^{21}$ a positive or negative number?

Summary!

We can raise a negative number to a power too.

$(-5)^2$ for example

Use brackets to show that the negative is part of the number (we calculate indices first in BIDMAS).

$(-5)^2=(-5)\times (-5)=25$

$-5^2=-(5\times 5)=-25$

When the base number is negative, the power will determine whether the answer is positive or negative.

Negative base number + **EVEN** power = **Positive number**
Negative base number + **ODD** power = **Negative number**