Video Transcript
Three squared plus one divided by
four squared plus two over three times four minus three squared minus one is equal
to what.
In this question, we are asked to
evaluate a given expression. To do this, we first need to recall
the order of operations. We can do this by using the acronym
PEMDAS. This tells us that we first perform
operations inside parentheses, then any exponents followed by multiplication, then
division. After this, we calculate additions,
and finally we evaluate the subtractions. It is also worth noting that we can
evaluate multiplication and division in any order. And we can also evaluate addition
and subtraction in any order. So we often group each pair of
operations in the same step.
To apply this to our expression, we
first need to start with the expression inside the parentheses. We can see that this expression
contains fraction notation. And before we evaluate this, we
need to recall that this notation means that we need to divide the entire numerator
by the entire denominator. So, we want to evaluate the
numerators and denominators separately. We can think of this as adding
parentheses around these expressions as shown. However, we usually leave the
parentheses out for visibility.
We now move on to evaluating any
exponents inside the parentheses. We can see that there are three
exponents inside the parentheses that we want to evaluate. We can calculate that three squared
is three times three, which is equal to nine. And four squared is four times
four, which is equal to 16. Therefore, our expression
simplifies to give us nine plus one divided by 16 plus two over three times four
minus nine minus one.
Let’s now move on to evaluating the
multiplications and divisions inside the parentheses, where we note that this does
not include the fraction notation with unsimplified numerators and denominators. There is only a single
multiplication or division that we can evaluate, and that is three times four, which
is equal to 12. This gives us nine plus one divided
by 16 plus two over 12 minus nine minus one.
We can now evaluate any additions
and subtractions in the numerators and denominators inside the parentheses. We can calculate that nine plus one
is equal to 10 and 12 minus nine is equal to three. This gives us 10 over 16 plus two
over three minus one.
Now that we have evaluated each of
the numerators and denominators inside the parentheses, we can apply the order of
operations once again to evaluate. We see that there are no exponents
to evaluate. We can then evaluate any
multiplications or divisions. This means that we can cancel the
shared factor of two in the numerator and denominator of the first fraction inside
the parentheses to obtain five over eight.
We then need to evaluate the
addition. And to do this, we need the
denominators of the two fractions to be equal. Since the lowest common multiple of
the denominators is 24, we multiply the first fraction by three over three and the
second fraction by eight over eight to obtain 15 over 24 plus 16 over 24 minus
one. We can then add the fractions by
adding their numerators. We get 31 over 24, and we need to
subtract one from this fraction.
We can now evaluate operations
outside of the parentheses. We cannot simplify the fraction, so
we will evaluate the subtraction. To do this, we need their
denominators to be equal, so we rewrite one as 24 over 24. This allows us to subtract the
fractions by subtracting their numerators, giving us seven over 24. We cannot simplify this any
further, so our final answer is seven over 24.
