Part a) Calculate one-sixth plus
Now, the key rule that we need to
remember when adding fractions is that they need to have the same denominator. So that’s the same number on the
bottom of the fraction. To make our life as easy as
possible, this number should be the lowest common multiple. So that’s the smallest number that
appears in both the times tables of the two denominators.
So in this case, we want the lowest
common multiple of six and five. The lowest common multiple of six
and five is 30, which is actually equal to six multiplied by five. But that won’t always be the
case. We need to convert both one-sixth
and three-fifths into equivalent fractions, where the denominators are 30. Now in order for these fractions to
be equivalent to the original fractions, we need to make sure we’ve done the same
thing to both the numerator and the denominator.
So to get from six to 30, we’ve
multiplied by five. Which means we also need to
multiply the one in the numerator by five to give the numerator in the equivalent
fraction. One multiplied by five is five. So this tells us that the fraction
one-sixth is equivalent to five thirtieths or five over 30. For our second fraction,
three-fifths, we’ve had to multiply five by six in order to give 30. So we need to do the same thing in
the numerator. Three multiplied by six is 18. So the fraction three-fifths is
equivalent to 18 over 30. Now, we have the same denominators
for the two fractions. We can add them. So one-sixth plus three-fifths
becomes five over 30 plus 18 over 30.
The next key thing we need to
remember is that if we’re adding two fractions, which now have the same denominator,
the denominator stays the same and we add the numerators. This is because if you have
five-thirtieths and you add another eighteen thirtieths, in total, you have five
plus eighteen thirtieths. So this is the fraction five plus
18 over 30. Five plus 18 is 23. So this gives the fraction 23 over
30 or twenty-three thirtieths.
We should always check whether a
fraction can be simplified by looking for common factors in the numerator and
denominator. Here, 23 is a prime number and it
isn’t a factor of 30. So this fraction can’t be
simplified any further.
Part b) Find the value of four to
the power of negative two.
Now in this question, we have a
negative power. So we need to recall what this
means. In fact, a negative power describes
a reciprocal, which means one divided by a value. A general rule is that 𝑥 to the
power of negative 𝑚 is equal to one divided by 𝑥 to the power of 𝑚. So we change the sign of the power
from negative to positive. But we do one divided by this
value. So this means that four to the
power of negative two is equal to one divided by four to the power of two or four
Now, we can evaluate the
denominator of this fraction. Four squared, remember, means four
multiplied by four which is equal to 16. So one over four squared is equal
to one over 16. The value of four to the power of
negative two is one over 16 or one sixteenth. It’s perfect okay to leave this
value as a fraction. We don’t need to try and convert it
into a decimal.
Now, just a quick word about common
mistakes that students often make. The first one is to think that four
to the power of negative two means four multiplied by negative two, which it
doesn’t. The second is to think that this
negative power means the answer will be negative. So to think that four squared is 16
means that four to power of negative two must be negative 16. Again, this is incorrect. You need to remember the rule that
a negative power means we’ll be working out a reciprocal.