Question Video: Factorizing the Sum of Two Cubes

Factorise fully 𝑎²⁴ + 𝑏²⁷.

02:29

Video Transcript

Factorise fully 𝑎 to the power of 24 plus 𝑏 to the power of 27.

We can answer this question by using the fact that the sum of two cubes, 𝑎 cubed plus 𝑏 cubed, is equal to 𝑎 plus 𝑏 multiplied by 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared. Using one of our laws of indices or exponents, we can rewrite 𝑎 to the power of 24 and 𝑏 to the power of 27.

𝑎 to the power of 24 can be rewritten as 𝑎 to the power of eight cubed. And 𝑏 to the power of 27 can be rewritten as 𝑏 to the power of nine cubed. This is because eight multiplied by three is equal to 24. And nine multiplied by three is equal to 27. This means that we can rewrite 𝑎 to the power of 24 plus 𝑏 to the power of 27 as 𝑎 to the power of eight cubed plus 𝑏 to the power of nine cubed.

We now have the sum of two cubes. Using our rule, the first bracket or parenthesis will become 𝑎 to the power of eight plus 𝑏 to the power of nine. The second bracket will become 𝑎 to the power of eight squared minus 𝑎 to the power of eight multiplied by 𝑏 to the power of nine plus 𝑏 to the power of nine squared. Once again, we can use our laws of exponents to simplify the first and third term in this parenthesis. Eight multiplied by two is equal to 16. And nine multiplied by two is equal to 18.

This means that the full factorisation of 𝑎 to the power of 24 plus 𝑏 to the power of 27 is 𝑎 to the power of eight plus 𝑏 to the power of nine multiplied by 𝑎 to the power of 16 minus 𝑎 to the power of eight 𝑏 to the power of nine plus 𝑏 to the power of 18. We could check this answer by expanding or multiplying out the parentheses.

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