Video Transcript
Calculate negative three and a
fifth to the seventh power times negative one and a half to the sixth power over
negative 16 over five to the sixth power times negative three over two to the fourth
power, giving your answer in its simplest form.
Although this question can look
quite difficult, we’re going to start by writing the mixed numbers as improper
fractions and then see what exponent laws we could apply. Let’s start with the mixed number
negative three and one-fifth. The whole number part of three is
made up of three five-fifths plus one-fifth left over would give us sixteen
fifths. We weren’t dealing with just three
and a fifth; it was negative three and a fifth. So we’ll have negative 16 over five
to the seventh power.
Next, negative one and a half is
equivalent to negative three over two and that’s to the sixth power. We can keep the denominator as it
was as our fractions are improper here and not mixed members. At this point, we might hopefully
begin to notice something about what we’ve written. We can see that we have a negative
16 over five on the numerator and denominator. And we also have a negative three
over two on the numerator and denominator. At this point, we can start to
think if we can cancel on the top and the bottom of this fraction.
Let’s consider the first part of
this fraction. We can use the quotient exponent
law 𝑥 to the power of 𝑎 over 𝑥 to the power of 𝑏 equals 𝑥 to the power of 𝑎
minus 𝑏, because we have the same value, which is taken to a power. That’s our value of 𝑥. Our answer will have our 𝑥. That’s negative 16 over five raised
to a power. To work out this power, we have
seven take away six, which gives us one. Negative 16 over five to the power
of one is the same as negative 16 over five.
Now, let’s simplify the second part
of this fraction. Using the same rule of exponents to
simplify negative three over two to the sixth power over negative three over two to
the fourth power gives us negative three over two squared, remembering that the
squared comes from six subtract four. So now, we have a simplified
calculation, negative 16 over five times negative three over two squared. If we look at negative three over
two squared, this is equivalent to negative three squared on the numerator and two
squared on the denominator. This is because we can apply the
exponent law 𝑥 over 𝑦 to the power of 𝑎 equals 𝑥 to the power of 𝑎 over 𝑦 to
the power of 𝑎.
So let’s simplify what we can in
our calculation. Negative three squared is nine and
two squared is four. We can multiply fractions by
multiplying the numerators and denominators but before that observe that we can
simplify this. Four is a common factor of both
negative 16 and four. So we calculate negative four times
nine, which is negative 36. And five times one gives us
five. So our final answer is negative 36
over five. It would also have been valid to
give our answer as negative seven and one-fifth.
