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
explanation: The document is an educational text that explains how to handle indices with negative numbers. It covers the rules for squaring and cubing negative numbers and the general pattern for raising negative numbers to even and odd powers. The text includes several questions and answers to guide the reader through the concepts.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an educational text that explains how to handle indices with negative numbers. It covers the rules for squaring and cubing negative numbers and the general pattern for raising negative numbers to even and odd powers. The text includes several questions and answers to guide the reader through the concepts.
  
  Label: multiple-question-and-answer