Given that 𝑥 is equal to
three-quarters and 𝑦 is equal to negative two-fifths, find the value of 𝑥 plus 𝑦
In this question, we are given the
values of two rational numbers 𝑥 and 𝑦 and asked to use these values to evaluate
the expression 𝑥 plus 𝑦 all squared.
To do this, we first need to
substitute the given values of 𝑥 and 𝑦 into the expression. This gives us three-quarters plus
negative two-fifths all squared.
We can recall the order of
operations by using the acronym PEMDAS. The P means that we start with
expressions inside the parentheses. And we can see that we have
three-quarters plus negative two-fifths inside the parentheses. We can then recall that adding a
negative number is the same as subtracting the positive of that number. So we can rewrite the expression as
three-quarters minus two-fifths all squared.
We still need to evaluate the
operation inside the parentheses. And since this is a subtraction of
rational numbers, we need them to have the same denominator. We can note that the lowest common
multiple of the denominators is 20. We can rewrite three-quarters as 15
over 20 by multiplying its numerator and denominator by five. And we can rewrite two-fifths as
eight over 20 by multiplying its numerator and denominator by four. This gives us 15 over 20 minus
eight over 20 all squared.
We can now evaluate the subtraction
by recalling that we can subtract fractions with the same denominator by finding the
difference in their numerators. In general, we have that 𝑎 over 𝑐
minus 𝑏 over 𝑐 is equal to 𝑎 minus 𝑏 all over 𝑐, provided that 𝑐 is
nonzero. Since 15 minus eight is equal to
seven, our expression simplifies to be seven over 20 all squared.
We now need to evaluate the square
of a rational number. And we can do this by recalling
that we can square a rational number by squaring its numerator and denominator
separately. In general, we have that 𝑎 over 𝑏
all squared is equal to 𝑎 squared over 𝑏 squared provided that 𝑏 is nonzero. Applying this result with 𝑎 equal
to seven and 𝑏 equal to 20 gives us seven squared over 20 squared.
Finally, we can calculate that
seven squared is seven times seven, which is equal to 49, and 20 squared is 20 times
20, which is equal to 400, giving us 49 over 400. We cannot simplify this any
further. So we have shown that 𝑥 plus 𝑦
all squared will be equal to 49 over 400.