text: Albert Teen
  
  YOU ARE LEARNING:
  
  Indices: Negative Numbers
  
  # Indices: Negative Numbers
  
  ### Indices: Negative Numbers
  
  We can have negative numbers as the base number and must be able to calculate indices with these.
  
  1
  
  An index number is a base number raised to a power. How do you write $6$ squared?
  
  2
  
  We can also raise negative numbers to a power. What is the correct way of writing minus $3$ squared?
  
  3
  
  Brackets are needed to be clear that we are squaring the negative number.
  
  Without brackets, we square the number first and then make negative.
  
  4
  
  What is another way of writing $(-3)^2$?
  
  5
  
  What is $(-3)^2$?
  
  Work out $(-6)^2$
  
  1
  
  There is a pattern when squaring negative numbers. What type of number is the answer?
  
  2
  
  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 $(-3)^3$
  
  1
  
  Cubing multiplies a number by itself three times
  
  So $(-3)^3$ is the same as $(-3) \times (-3) \times (-3)$
  
  2
  
  What is $(-3) \times (-3)$?
  
  3
  
  What is $9 \times (-3)$?
  
  4
  
  Cubing negative numbers always returns a negative
  
  This is true for every negative number. Nice 👍
  
  What is $(-5)^3$?
  
  1
  
  We have seen that
  
  Squaring a negative number gives a positive answer Cubing a negative number gives a negative answer
  
  2
  
  What about other powers? What is $(-3)^4$?
  
  3
  
  The answer to $(-3)^4$ was positive. Will the answer to $(-3)^5$ be positive or negative?
  
  4
  
  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!
  
  1
  
  We can raise a negative number to a power too.
  
  $(-5)^2$ for example
  
  2
  
  Use brackets to show that the negative is part of the number (we calculate indices first in BIDMAS).
  
  $(-5)^2=(-5)\times (-5)=25$
  
  $-5^2=-(5\times 5)=-25$
  
  3
  
  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