Video Transcript
What is the additive inverse of the square root of 65 minus the square root of 85?
In this question, we are asked to find the additive inverse of a given real number. We should begin by recalling what is meant by the additive inverse of a real number. We recall that if 𝑎 plus 𝑏 is equal to zero, then we say that 𝑏 is the additive inverse of 𝑎, because their sum is the additive identity. This is the definition of a general additive inverse.
However, when working with real numbers, we know that 𝑎 plus negative 𝑎 is equal to zero. So, negative 𝑎 is the additive inverse of 𝑎. This means that we can find the additive inverse of any real number by multiplying it by negative one. Hence, the additive inverse to the square root of 65 minus the square root of 85 is negative one multiplied by the square root of 65 minus the square root of 85. We can simplify this expression for the additive inverse by distributing the factor of negative one over the terms inside the parentheses. This gives us negative the square root of 65 plus the square root of 85. We can then reorder the two terms to obtain the square root of 85 minus the square root of 65.
It is worth noting that we can check our answer by checking that 𝑎 plus negative 𝑎 is equal to zero. Substituting the values of 𝑎 and negative 𝑎 into the expression gives us root 65 minus root 85 plus root 85 minus root 65. We can then see that negative root 85 plus root 85 is zero and root 65 minus root 65 is zero, confirming that the square root of 85 minus the square root of 65 is the additive inverse to root 65 minus root 85.
