Difference between revisions of "Perfect power"
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| − | A [[positive integer]] <math>n</math> is a '''perfect power''' if there | + | A [[positive integer]] <math>n</math> is a '''perfect power''' if there exist integers <math>m, k</math> such that <math>k \geq 2</math> and <math>n = m^k</math>. In particular, <math>n</math> is said to be a ''perfect <math>k</math>th power''. For example, <math>64 = 8^2 = 4^3 = 2^6</math>, so <math>64</math> is a perfect <math>2</math>nd, <math>3</math>rd and <math>6</math>th power. |
| − | We restrict <math>k \geq 2</math> only because "being a perfect | + | We restrict <math>k \geq 2</math> only because "being a perfect <math>1</math>st power" is a meaningless property: every integer is a <math>1</math>st power of itself. |
| − | + | Perfect second powers are also known as [[perfect square]]s and perfect third powers are also known as [[perfect cube]]s. This is because the [[area]] of a [[square (geometry) | square]] and the [[volume]] of a [[cube (geometry) | cube]] is equal to the second and third powers of a side length, respectively. | |
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| − | Perfect second powers are | ||
Latest revision as of 16:52, 18 August 2013
A positive integer 
is a perfect power if there exist integers 
such that 
and 
. In particular, is said to be a perfect 
th power. For example, 
, so 
is a perfect 
nd, 
rd and 
th power.
We restrict only because "being a perfect 
st power" is a meaningless property: every integer is a st power of itself.
Perfect second powers are also known as perfect squares and perfect third powers are also known as perfect cubes. This is because the area of a square and the volume of a cube is equal to the second and third powers of a side length, respectively.
