Powers, Roots & Indices (OCR GCSE Maths)

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Powers & Roots

What are powers/indices?

  • Powers of a number is when that number is multiplied by itself repeatedly
    • 51 means 5
    • 52 means 5 × 5
    • 53 means 5 × 5 × 5
    • Therefore the powers of 5 are 5, 25, 125, etc
  • The big number on the bottom is sometimes called the base number
  • The small number that is raised is called the index or the exponent
  • Any non-zero number to the power of 0 is equal to 1
    • 50 = 1

What are roots?

  • Roots of a number are the opposite of powers
  • A square root of 25 is a number that when squared equals 25
    • The two square roots are 5 and -5 
    • Every positive number has two square roots
      • They will have the same digits but one is positive and one is negative
    • The notation square root of blank end root  refers to the positive square root of a number
      • square root of 25 equals 5
      • You can show both roots at once using the plus or minus symbol ±
      • Square roots of 25 are plus-or-minus square root of 25 equals plus-or-minus 5
    • The square root of a negative number is not a real number
    • The positive square root can be written as an index of 1 half so 25 to the power of 1 half end exponent equals 5
  • cube root of 125 is a number that when cubed equals 125
    • A cube root of 125 is 5
    • Every positive and negative number always has a cube root
    • The notation cube root of blank refers to the cube root of a number
      • cube root of 125 equals 5
    • The cube root can be written as an index of 1 third so 125 to the power of 1 third end exponent equals 5
  • A nth root of a number (n-th root of blank)is a number that when raised to the power n equals the original number
    • If n is even then they work the same way as square roots
      • Every positive number will have a positive and negative nth root
      • The notation n-th root of blank refers to the positive nth root of a number
    • If n is odd then they work the same way as cube roots
      • Every positive and negative number will have an nth root
    • The nth root can be written as an index of 1 over n
  • If you know your powers of numbers then you can use them to find roots of numbers
    • e.g. 2 to the power of 5 equals 32 means fifth root of 32 equals 2
      • You could write this using an index 32 to the power of 1 fifth end exponent equals 2
  • You can also estimate roots by finding the closest powers
    • e.g. 2 cubed equals 8 and 3 cubed equals 27 therefore 2 less than cube root of 20 less than 3

What are reciprocals?

  • The reciprocal of a number is the number that you multiply it by to get 1
    • The reciprocal of 2 is 1 half
    • The reciprocal of 0.25 or 1 fourth is 4
    • The reciprocal of 3 over 2 is 2 over 3
  • The reciprocal of a number can be written as a power with an index of -1
    • 5-1 means the reciprocal of 5
  • This idea can be extended to other negative indices
    • 5-2 means the reciprocal of 52

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Laws of Indices

What are the laws of indices?

  • There are lots of very important laws (or rules)
  • It is important that you know and can apply these
  • Understanding the explanations will help you remember them
Law Description Why
a to the power of 1 equals a anything to the power 1 is itself 6 to the power of 1 equals 6
a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent to multiply indices with the same base, add their powers 4 cubed cross times 4 squared
equals open parentheses 4 cross times 4 cross times 4 close parentheses cross times open parentheses 4 cross times 4 close parentheses
equals 4 to the power of 5
a to the power of m divided by a to the power of n equals a to the power of m over a to the power of n equals a to the power of m minus n end exponent to divide indices with the same base, subtract their powers 7 to the power of 5 divided by 7 cubed
equals fraction numerator 7 cross times 7 cross times up diagonal strike 7 cross times up diagonal strike 7 cross times up diagonal strike 7 over denominator up diagonal strike 7 cross times up diagonal strike 7 cross times up diagonal strike 7 end fraction
equals 7 squared
amn=amn{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} to raise indices to a new power, multiply their powers 1432=14×14×14×14×14×14=146{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}
a0=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} anything to the power 0 is 1 80=82-2=82÷82=8282=1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}
a-n=1an{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} a negative power is "1 over" the positive power 11-3=110-3=110÷113=110113=1113{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}
a1n=an{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} a power of an nth is an nth root 5122=512×2=51=5so   512=5{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}
amn=anm=amn {"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

a fractional power of m over n means either

- do the the nth root first, then raise it to the power m

or

- raise it to the power m, then take the nth root

(depending on what's easier)

932=912×3=9123=93or  932=93×12=9312=93{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}
abn=anbn{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} a power outside a fraction applies to both the numerator and the denominator 563 =56×56×56=5363{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}
ab-n=ban=bnan{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} flipping the fraction inside changes a negative power into a positive power 56-2=1562=1÷562=1÷5262=1×6252=6252=652{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

How do I apply more than one of the laws of indices?

  • Powers can include negatives and fractions
    • These can be dealt with in any order
    • However the following order is easiest as it avoids large numbers
  • If there is a negative sign in the power then deal with that first
    • Take the reciprocal of the base number
    • open parentheses 8 over 27 close parentheses to the power of negative 2 over 3 end exponent equals open parentheses 27 over 8 close parentheses to the power of 2 over 3 end exponent
  • Next deal with the denominator of the fraction of the power
    • Take the root of the base number
    • open parentheses 27 over 8 close parentheses to the power of 2 over 3 end exponent equals open parentheses fraction numerator cube root of 27 over denominator cube root of 8 end fraction close parentheses squared equals open parentheses 3 over 2 close parentheses squared
  • Finally deal with the numerator of the fraction of the power
    • Take the power of the base number
    • open parentheses 3 over 2 close parentheses squared equals 3 squared over 2 squared equals 9 over 4

How do I deal with different bases?

  • Sometimes expressions involve different base values
  • You can use index laws to change the base of a term to simplify an expression involving terms with different bases
    • For example begin mathsize 20px style 9 to the power of 4 equals left parenthesis 3 squared right parenthesis to the power of 4 equals 3 to the power of 2 cross times 4 end exponent equals 3 to the power of 8 end style
    • Using the above can then help with problems like 9 to the power of 4 divided by 3 to the power of 7 equals 3 to the power of 8 divided by 3 to the power of 7 equals 3 to the power of 8 minus 7 end exponent equals 3 to the power of 1 equals space 3

Examiner Tip

  • Index laws only work with terms that have the same base, so something like 23 × 52 cannot be simplified using index laws

Worked example

(a)table row cell blank to the power of blank end cell row blank end table
Without using a calculator, write fraction numerator 7 cubed cross times 7 squared over denominator 7 to the power of 8 end fraction in the form 1 over 7 to the power of k where k is a positive whole number.

Use a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent on the numerator.
 

7 cubed cross times 7 squared equals 7 to the power of 3 plus 2 end exponent equals 7 to the power of 5
 

Use a to the power of m divided by a to the power of n equals a to the power of m minus n end exponent
 

7 to the power of 5 over 7 to the power of 8 equals 7 to the power of 5 minus 8 end exponent equals 7 to the power of negative 3 end exponent
 

Use a to the power of negative n end exponent equals 1 over a to the power of n.
 

7 to the power of negative 3 end exponent equals 1 over 7 cubed
 

The value of k is 3.

bold 1 over bold 7 to the power of bold 3 

(b)table row blank row blank end table
Without using a calculator, simplify open parentheses 4 over 25 close parentheses to the power of negative 3 over 2 end exponent.

Flip the fraction to change the negative outside power into a positive outside power, open parentheses a over b close parentheses to the power of negative n end exponent equals open parentheses b over a close parentheses to the power of n.
 

open parentheses 4 over 25 close parentheses to the power of negative 3 over 2 end exponent equals open parentheses 25 over 4 close parentheses to the power of 3 over 2 end exponent
 

Use that a power outside a fraction applies to both the numerator and denominator, open parentheses a over b close parentheses to the power of n equals a to the power of n over b to the power of n.
 

open parentheses 25 over 4 close parentheses to the power of 3 over 2 end exponent equals 25 to the power of begin display style 3 over 2 end style end exponent over 4 to the power of begin display style 3 over 2 end style end exponent
 

Use that a fractional power of m over n is the nth root all to the power m,  a to the power of m over n end exponent equals open parentheses n-th root of a close parentheses to the power of m.
 

25 to the power of 3 over 2 end exponent equals open parentheses square root of 25 close parentheses cubed equals 5 cubed equals 125  and   4 to the power of 3 over 2 end exponent equals open parentheses square root of 4 close parentheses cubed equals 2 cubed equals 8
 

bold 125 over bold 8

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