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If 𝑎³ − 𝑏³ = 45 and 𝑎 − 𝑏 = 9, what is the value of 𝑎² + 𝑎𝑏 + 𝑏²?

If 𝑎 cubed minus 𝑏 cubed is equal to 45 and 𝑎 minus 𝑏 is equal to nine, what is the value of 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared?

The difference of two cubes, 𝑎 cubed minus 𝑏 cubed, can be fully factorized as follows: 𝑎 minus 𝑏 multiplied by 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared. We were told that 𝑎 cubed minus 𝑏 cubed is equal to 45 and that 𝑎 minus 𝑏 is equal to nine. This means that 45 is equal to nine multiplied by 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared.

We can divide both sides of this equation by nine. 45 divided by nine is equal to five. Dividing the right-hand side of the equation by nine leaves us with 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared.

This means that if 𝑎 cubed minus 𝑏 cubed is equal to 45 and 𝑎 minus 𝑏 is equal to nine, then the value of 𝑎 squared plus 𝑎𝑏 plus 𝑏 squared is five.

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