Uses of Prime Factor Decomposition

Uses of Prime Factor Decomposition

Uses of prime factor decomposition

How can I use PFD to identify a square or cube number?

• If all the indices in the prime factor decomposition of a number are even, then that number is a square number

  • E.g. The prime factor decomposition of 7056 is 2⁴ × 3² × 7²

  • All powers are even so it must be a square number

    • It can be written as (2² × 3 × 7)²

• If all the indices in the prime factor decomposition of a number are multiples of 3, then that number is a cube number

  • E.g. The prime factor decomposition of 1728000 is 2⁹ × 3³ × 5³

  • All powers are multiples of 3 so it must be a cube number

    • It can be written as (2³ × 3 × 5)³

How can I use PFD to find the square root of a square number?

• Write the number in its prime factor decomposition

  • All the indices should be even if it is a square number

• For example, to find the square root of 144 = 2⁴ × 3²

  • Halve all of the indices

    • 2² × 3

    • So $\sqrt{2^4\times 3^2}=2^2\times 3$

• This is the prime factor decomposition of the square root of the number

  • To find it as an integer, multiply the prime factors together

  • 2² × 3 = 12, so the square root of 144 is 12

How can I use PFD to find the exact square root of a number?

• If the number is not a square number, its exact square root can still be found using its prime factor decomposition

• Write the number in its prime factor decomposition

  • $1440=2^5\times 3^2\times 5$

• Rewrite the prime factor decomposition with as many even indices as you can

  • E.g. 2³ = 2² × 2, or 5⁷ = 5⁶ × 5

  • $1440=2^4\times 2\times 3^2\times 5$

• Collect the terms with even powers together

  • $1440=2^4\times 3^2\times 2\times 5$

• Square root both sides

  • $\sqrt{1440}=\sqrt{2^4\times 3^2\times 2\times 5}$

• Using the rule $\sqrt{ab}=\sqrt{a}\sqrt{b}$, apply the square root to the terms with the even indices separately to the terms with odd indices

  • $\sqrt{1440}=\sqrt{2^4\times 3^2}\times \sqrt{2\times 5}$

• Simplify to find your answer, remembering that $\sqrt{a^2{b}^2}=ab$

  • $\sqrt{1440}=2^2\times 3\times \sqrt{10}$

  • $\sqrt{1440}=12\sqrt{10}$

  • $12\sqrt{10}$ is the exact square root of $1440$