Powers, Roots & Standard form

• What are powers/indices?

  Powers of a number are when that number is multiplied by itself repeatedly.
• How is $5^2$ expressed in multiplication form?


  $5^2 =$$5 \times 5$

  
• What does $5^3$ represent?


  $5^3 =$$5 \times 5 \times 5$
• What are the powers of 5?

  The powers of 5 are 5, 25, 125, etc.
• What is the big number in a power called?

  The big number is called the base number.
• What is the small number in a power called?

  The small number that is raised is called the index or the exponent.
• What is the value of any non-zero number raised to the power of 0?

  Any non-zero number to the power of 0 is equal to 1.
• What is a root of a number?

  Roots of a number are the opposite of powers.
• What is a square root of 25?

  A square root of 25 is a number that when squared equals 25, which are 5 and -5.
• How many square roots does every positive number have?

  Every positive number has two square roots.
• What does the notation $\sqrt{25}$ refer to?


  The notation $\sqrt{25}$ refers to the positive square root of a number.
• How can both square roots of 25 be shown at once?

  Both square roots can be shown using the plus or minus symbol $±$.
• What is the square root of a negative number?

  The square root of a negative number is not a real number.
• How can the positive square root be expressed as an index?


  The positive square root can be written as an index of $\frac{1}{2}$.
• What is a cube root of 125?

  A cube root of 125 is a number that when cubed equals 125, which is 5.
• How many cube roots does every positive and negative number have?

  Every positive and negative number always has a cube root.
• What does the notation $\sqrt[3]{125}$ refer to?


  The notation $\sqrt[3]{125}$ refers to the cube root of a number.
• How can a cube root be expressed as an index?

  A cube root can be written as an index of $\frac{1}{3}$.
• What is an nth root of a number?

  An nth root of a number is a number that when raised to the power n equals the original number.
• How do nth roots behave when n is even?

  If n is even, then they work the same way as square roots.
• How do nth roots behave when n is odd?

  If n is odd, then they work the same way as cube roots.
• How can the nth root be expressed as an index?

  The nth root can be written as an index of $\frac{1}{n}$.
• How can powers of numbers be used to find roots?

  If you know your powers of numbers, you can use them to find roots of numbers.
• What does $25=$$3^2$ mean?


  $25=$$3^2$ means $5^{3/2} =$$2$.
• How can roots be estimated?

  You can estimate roots by finding the closest powers.
• What is a reciprocal of a number?

  The reciprocal of a number is the number that you multiply it by to get 1.
• What is the reciprocal of 2?

  The reciprocal of 2 is $\frac{1}{2}$.
• What is the reciprocal of 0.25?

  The reciprocal of 0.25 or $\frac{1}{4}$ is 4.

  
• What is the reciprocal of $\frac{3}{2}$?


  The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$.
• How can the reciprocal of a number be expressed as a power?

  The reciprocal of a number can be written as a power with an index of -1.
• What does $5^{-1}$ represent?


  $5^{-1}$ means the reciprocal of 5.
• What does $5^{-2}$ represent?


  $5^{-2}$ means the reciprocal of $5^2$.
• What are the laws of indices?

  The laws of indices are important rules for manipulating powers.
• How do you apply more than one of the laws of indices?

  Powers can include negatives and fractions, and these can be dealt with in any order.
• What is the first step when dealing with a negative power?

  The first step is to take the reciprocal of the base number.
• What is the second step when dealing with a fraction in the power?

  The second step is to take the root of the base number.
• What is the final step when dealing with a fraction in the power?

  The final step is to take the power of the base number.
• How can different bases be dealt with in expressions?


  You can use index laws to change the base of a term to simplify an expression involving different bases.
• What is an example of changing bases?

  An example is $9^4 =$$(3^2)^4 =$$3^{2 \times 4} =$$3^8$.

  
• What is the limitation of index laws?

  Index laws only work with terms that have the same base.