Given that 𝑥 is equal to 29 minus root 50 and 𝑦 is the multiplicative inverse of 𝑥, determine the value of 𝑥 minus 𝑦 squared.
In order to answer this question, we will use the properties of conjugate irrational numbers. The irrational conjugate of an irrational number 𝑎 plus root 𝑏 is 𝑎 minus root 𝑏. In this question, we’re told that 𝑥 is equal to 29 minus root 50. When dealing with irrational numbers of this type, the multiplicative inverse is the irrational conjugate. This means that 𝑦 is equal to 29 plus root 50. By substituting in these values of 𝑥 and 𝑦, we need to calculate 𝑥 minus 𝑦 all squared. This means that we need to subtract 29 plus root 50 from 29 minus root 50 and then square the answer.
Distributing our parentheses, we get 29 minus root 50 minus 29 minus root 50 all squared. The 29s cancel, so we are left with negative two root 50 squared. In order to calculate this, we can square negative two and root 50 separately. Multiplying two negative numbers together gives a positive answer. So negative two squared is equal to four. When squaring a radical or surd root 𝑎, we are multiplying root 𝑎 by root 𝑎. This is equal to 𝑎. This means that root 50 squared is equal to 50. Finally, multiplying four by 50 gives us 200.
If 𝑥 is equal to 29 minus root 50 and 𝑦 is the multiplicative inverse of 𝑥, then 𝑥 minus 𝑦 all squared is equal to 200.