Erin is trying to expand the brackets in the expression two 𝑥 minus three times 𝑥
minus four. Her working is shown as follows: two 𝑥 minus three times 𝑥 minus four is equal to
two 𝑥 times 𝑥 minus four minus three times 𝑥 minus four which is equal to two 𝑥
squared minus eight 𝑥 minus three 𝑥 minus 12 which is equal to two 𝑥 squared
minus 11𝑥 minus 12. Part a) What mistake has she made in her working?
There will also be a part b. So Erin is trying to expand the brackets. So she needs to take two 𝑥 times 𝑥 and two 𝑥 times negative four. And this step is represented here. She will also need to take negative three times 𝑥 and negative three times negative
four which is shown here. So two 𝑥 times 𝑥 is two 𝑥 squared. Two 𝑥 times negative four is negative eight 𝑥. So she’s done well so far.
Negative three times 𝑥 is negative three 𝑥. However, negative three times negative four should be equal to positive 12. And here, she has a negative 12. Going from the second to the third step, she brought down the two 𝑥 squared,
combined negative eight 𝑥 and negative three 𝑥 to get negative 11𝑥. So that is correct.
However, once again, this negative 12 needs to be a positive 12. So the mistake that she made in her working was her multiplication of negative three
and negative four. It needed to be a positive 12. So her answer should have been two 𝑥 squared minus 11𝑥 plus 12. And making the mistake of writing negative 12 instead of positive 12 is common. Just remember when multiplying a negative number to a negative number, it turns into
a positive number.
Now, let’s move on to part b. Jacob is trying to simplify the expression eight 𝑦 cubed times three 𝑦 squared
divided by four 𝑦 to the power of negative three. His working is shown as follows: eight 𝑦 cubed times three 𝑦 squared divided by
four 𝑦 to the power of negative three is equal to 24𝑦 to the power of five divided
by four 𝑦 to the power of negative three which is equal to 24𝑦 squared divided by
four which is equal to six 𝑦 squared. What mistake has he made in his working?
Notice in Jacob’s first step. He has taken eight 𝑦 cubed times three 𝑦 squared to get 24𝑦 to the power of five
because we would need to multiply eight and three together to get 24 which is
correct. So here, we need to add our powers together because we have like bases and we’re
multiplying. So this needs to be 𝑦 to the three plus two power which would be 𝑦 to the power of
five which is what he has. So so far, Jacob has done this correctly.
Now, for the next part, he still has his 24 divided by the four. So he has not worked with these numbers yet, which means he’s decided to simplify 𝑦
to the power of five divided by 𝑦 to the power of negative three. And when we divide, when we have like bases, we actually need to subtract our
powers. So we need five minus negative three.
Now, this is a very common place to make a mistake. Five minus negative three should actually be five plus three. That’s what the two minus signs turn into: a plus sign. And five plus three is eight. So the 𝑦 squared should really be a 𝑦 to the power of eight.
So Jacob made this common mistake of seeing the minus sign and taking five minus three
to get two 𝑦 squared. But it should be 𝑦 to the power of eight. And then, if we will completely simplify and finish this question, 24𝑦 to the power
of eight divided by four means we need to take 24 divided by four to get six and
then we also have our 𝑦 to the power of eight.
So what mistake has he made in his working? That was here with the 𝑦 squared. As we said before, 𝑦 to the power of five divided by 𝑦 to the power of negative
three should be 𝑦 to the power of five minus negative three which is 𝑦 to the
power of five plus three which is 𝑦 to the power of eight. So the answer should be six 𝑦 to the power of eight.