Video Transcript
Given that 𝑎 is equal to the square root of two plus nine and 𝑏 is equal to the square root of two minus nine, find the value of 𝑎 minus 𝑏.
In this question, we are given the values of 𝑎 and 𝑏 and asked to evaluate the expression 𝑎 minus 𝑏.
To evaluate this expression, we need to start by substituting the values of 𝑎 and 𝑏 into the expression. This gives us the square root of two plus nine minus the square root of two minus nine. To simplify the expression further, we can distribute the negative over the terms inside the parentheses by using the distributive property of the multiplication of real numbers over subtraction. This tells us that for any real numbers 𝑐, 𝑑, and 𝑒, we have 𝑐 multiplied by 𝑑 minus 𝑒 is equal to 𝑐 times 𝑑 minus 𝑐 times 𝑒.
We know that subtracting a number is the same as adding that number multiplied by negative one. So we can set 𝑐 equal to negative one, 𝑑 equal to the square root of two, and 𝑒 equal to nine. We get the square root of two plus nine minus the square root of two plus nine. We can then evaluate the sum of these real numbers in any order. And we can reorder the terms of the sum. We note that root two and negative root two are additive inverses. So they add to give zero.
Hence, we have shown that 𝑎 minus 𝑏 is equal to nine plus nine, which is equal to 18.
