Video Transcript
In this video, we will learn how to
apply the laws of exponents to multiply and divide powers and also how to work out a
power raised to a power. We will recap exponents and go
through some questions, including a question that looks like this. Using the laws of exponents, we’ll
see how questions like this can be made so much more manageable. Let’s begin by recalling the basics
of exponents or powers.
The small number here is often
referred to as the index or exponent. Three raised to the fourth power or
three to the power of four means that we have four threes all multiplied
together. We could calculate the value of
this in several ways. We could begin by multiplying three
times three, which is nine. Multiplying nine by another three
would give us 27. And then multiplying 27 by the
final three would give us 81. Alternatively, we could’ve got our
three times three, which is nine, and multiplied by the second lot of three times
three, which is nine again. And nine multiplied by nine is
81. So let’s take a look at what
happens when we multiply two values with exponents.
Fill in the blank: Negative two to
the seventh power times negative two to the fifth power equals negative two to the
power of what.
In this question, we’re asked to
find the index or power that negative two in the answer would be raised to. We can start by thinking about what
the value of negative two to the seventh power would look like. This would mean that we would have
negative two seven times. And all of these would be
multiplied together. In the same way, negative two to
the fifth power means that we have five negative twos all multiplied together. And our negative two to the seventh
power is multiplied by our negative two to the fifth power. So that means our seven negative
twos are multiplied by our five negative twos.
So how many negative twos have we
got in total now? Well, now, we have 12, which means
that we could write this as negative two raised to the power of 12. And so our missing exponent value
is 12. We could also have answered this
using the exponent rule that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of
𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. The value of 𝑥 in the original
question would be negative two; our two exponents 𝑎 and 𝑏 are seven and five. So our exponent value at the end is
equal to 𝑎 plus 𝑏 which would be seven plus five giving us 12, confirming our
original answer.
In this question, we saw our first
exponent rule that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is
equal to 𝑥 to the power of 𝑎 plus 𝑏. We know this to be true because if
we write our first 𝑥 to the power of 𝑎 as 𝑎 lots of 𝑥 multiplied together and
our second value of 𝑥 to the power of 𝑏 as 𝑏 lots of 𝑥, then when we add these
together, we know that we will have a total of 𝑎 plus 𝑏 lots of 𝑥. As these will all be multiplied
together, we can write this as 𝑥 to the power of 𝑎 plus 𝑏.
In the next question, we’ll have a
look at dividing values with exponents.
Is three to the 74th power over
three to the 75th power greater than, less than, or equal to three?
Let’s begin this question by
thinking about what three to the 74th power and three to the 75th power will
actually involve. We can recall that, for example,
three to the fourth power means that we write three down four times and multiply it
together. So three to the 74th power means
that there would be 74 lots of three multiplied together. Three to the 75th power would be 75
lots of three multiplied together. Actually, calculating the value of
either of these would take a very long time. And in this question, we also need
to divide. But let’s have a look at what
happens if we cancel values from the numerator and denominator.
We could cancel off the first
threes and the second two threes and so on until we have canceled off 74 threes on
the numerator and the denominator, leaving us with just three on the
denominator. We know that this will be
equivalent to one-third. So three to the 74th power over
three to the 75th power is equal to one-third. There is another way we could’ve
calculated this, though. The quotient law of exponents tells
us that 𝑥 to the power of 𝑎 over 𝑥 to the power of 𝑏 is equal to 𝑥 to the power
of 𝑎 minus 𝑏.
Here, the value of 𝑥 would be
three and our two exponents 𝑎 and 𝑏 would be 74 and 75. So our answer here would be three
to the power of 74 minus 75. And that’s equivalent to three to
the power of negative one. Three to the power of negative one
is the reciprocal of three, and that’s the same as one-third. Our question asks if this is
greater than, less than, or equal to three. One-third is less than three. And so that’s our answer.
We can pause for a second and add
this second quotient law to our laws of exponents. 𝑥 to the power of 𝑎 over 𝑥 to
the power of 𝑏 equals 𝑥 to the power of 𝑎 minus 𝑏. We can, of course, see the
equivalent question written with the division sign instead, 𝑥 to the power of 𝑎
divided by 𝑥 to the power of 𝑏. The value of this would still be
the same. So far, we’ve seen what happens
when we multiply values with exponents and divide values with exponents. But what happens when we take a
power of a power? Let’s find out in the next
question.
Find the value of negative three to
the third power times three squared to the third power.
In this question, we can see that
our values here are values that have exponents or powers. It’s worth noting that the wording
of “find the value” means that we’re not looking for an answer that has an
exponent. For example, if we found that this
was three to the third power, then we’d need to give the answer as 27 instead. We can recall that negative three
to the third power means that we have three lots of negative three multiplied
together. When we’re looking at three squared
to the third power, we can recall that three squared means three times three. And so writing this to the third
power means that we have three times three three times.
This is all becoming rather
complicated to say. So let’s compare this with the rule
that we have for finding the power of a power. In this case, 𝑥 to the power of 𝑎
to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏. If we look at three squared to the
third power, our value of three here is our 𝑥-value. So our answer will be three to the
power of two times three, three to the power of six. And we did indeed write six lots of
three and we multiplied together. So let’s work out our value when we
have negative three to the third power multiplied by three squared to the third
power or three to the power of six.
Let’s start with negative three
times negative three. We know that two negative values
multiplied together will give us a positive value, so we’ll start with nine. We then multiply by the next
negative three and nine times negative three is negative 27. Continuing to multiply by three, we
have negative 27 times three, which is negative 81. We can continue to multiply by
three bringing down our threes as we go until we get to a final answer.
It’s worth noting that there are,
of course, a number of ways in which we could’ve multiplied. We know that negative three times
negative three times negative three is negative 27 and three times three times three
is 27. And so multiplying negative 27 by
27 would give us negative 729, and multiplying that by the final 27 would also give
us the value of negative 19683. And this will be our final answer
for the value of negative three to the third power times three squared to the third
power.
Let’s pause for a minute to update
our notes. We saw that we can take the power
of a power, such as 𝑥 to the power of 𝑎 to the power of 𝑏, by multiplying the
exponents to give us 𝑥 to the power of 𝑎𝑏. So far in this video, we’ve been
looking at integers raised to a power. Now, let’s have a look at some
fractions raised to a power.
Which of the following is equal to
negative one and a half to the third power times negative one and a half
squared? Option (A) seven and 19 over 32,
option (B) negative seven and 19 over 32, option (C) 32 over 243, option (D)
negative 3125 over 1024, or option (E) negative 243.
In this question, we’re asked to
multiply these two values with exponents. When we’re working with fractions
and writing these with exponents, it’s always a good idea to make sure that the
fractions are written as top-heavy or improper fractions rather than a mixed
number. In both these cases, we have
negative one and a half. And so we can write this as
negative three over two to the third power times negative three over two
squared.
It’s useful to remember the product
law of exponents when we’re multiplying a value 𝑥 to the power of 𝑎 by 𝑥 to the
power of 𝑏. Then we add the exponents to give
us a value of 𝑥 to the power of 𝑎 plus 𝑏. In this case, our 𝑥-value will be
negative three over two and our exponent can be calculated by adding our indices
three and two, giving us five. So we have negative three over two
to the power of five. What we’ve done then is simplified
our calculation to give a value with an exponent. However, if we look at the answer
options, what we’re looking for here is to actually find the value of negative three
over two to the fifth power.
Breaking down what it actually
means to have negative three over two to the fifth power, that means we have five
lots of negative three over two all multiplied together. As a quick aside, when we’re
working with a fraction like negative three over two, there are several ways in
which we can write this, firstly, with the negative sign in front of the fraction or
with the negative three on the numerator or with the negative on the bottom to make
negative two. Although the final form is
mathematically correct, we tend to either have the negative sign on the numerator or
externally to the fraction.
Returning to the problem then, we
could think of this as a giant fraction with five lots of negative three multiplied
on the numerator and five lots of two multiplied on the denominator. So calculating the value then,
negative three times negative three is nine. Another negative three times
another negative three is nine. Nine times nine is 81. And 81 times negative three gives
us negative 243 on the numerator. On the denominator, we know that
two times two is four, four times four is 16, and multiplied by another two gives us
32 on the denominator.
We now need to write our improper
fraction of negative 243 over 32 as a mixed number. When we perform some long division
of 243 divided by 32, we get the value of seven with a remainder of 19. The value of seven represents the
whole number part of our answer. The remainder forms the numerator
of our fraction. And as we’ve divided by 32, then
that will be our denominator. We mustn’t forget that we were
dividing negative 243 by 32. So our value will be negative seven
and 19 over 32. We can see that this is the answer
given in option (B).
We’ll now look at one final
question involving exponents.
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.
We can now summarize what we’ve
learned in this video. We saw that there are a number of
laws we can use when working with exponents. The first two laws that we saw
covered what happens when we multiply values with exponents and when we divide
values with exponents. We saw the law covering what
happens when we take the power of a power. In this case, we would multiply the
exponents.
The final law that we saw covers
what happens when we have a fraction raised to the power of 𝑎. This would be equivalent to the
numerator and denominator both raised to the power of 𝑎. And one final point is to remember
that if we’re working with mixed numbers raised to a power, then always make sure we
change those to improper fractions.
