Video Transcript
In this video, we will learn how to
apply the laws of exponents to multiply and divide powers and to calculate a power
raised to a power over the real numbers. We should already be familiar with
what powers are and the key laws of exponents. But we we’ll recap them briefly
here.
A power is an expression of the
form 𝑎 to the power of 𝑚, where 𝑎 and 𝑚 are each real numbers. 𝑎 is called the base and 𝑚 is
called the exponent. If we’re able to evaluate this
power, so if we can write 𝑎 to the power of 𝑚 is equal to 𝑐, then 𝑐 is the
result. For example, two cubed or two to
the third power is equal to eight. For positive integer exponents,
such as the exponent of three we have here, we can think of powers as the number of
bases that we multiply together. So two cubed means two times two
times two; it’s three twos multiplied together.
There are several laws of exponents
which we should already be familiar with. Each of these apply for real values
of the base 𝑎. And for now, we’ll just be
concerned with integer values of the exponents. First, we have the product rule for
exponents, which tells us how to find the exponent when we have a product of powers
of the same base. We add the powers. 𝑎 to the power of 𝑚 multiplied by
𝑎 to the power of 𝑛 is 𝑎 to the power of 𝑚 plus 𝑛. So for example, seven squared
multiplied by seven cubed is seven to the power of two plus three; that’s seven to
the power of five. We can demonstrate this rule at
least for positive integer exponents by writing the expression out longhand. 𝑎 to the power of 𝑚 multiplied by
𝑎 to the power of 𝑛 is 𝑚 lots of 𝑎 multiplied together multiplied by 𝑛 lots of
𝑎 multiplied together. So overall, we’re multiplying 𝑚
plus 𝑛 lots of 𝑎 together. And so this is the exponent for the
result.
We do need to be careful when
working with this rule because there are two really common mistakes. The first is to multiply rather
than add the exponents. The second is to multiply the bases
or indeed make both mistakes and multiply both the bases and the exponents. But if we remember why this rule
holds, then we shouldn’t make these mistakes. Now, even though we demonstrated
the logic behind this rule using positive integer exponents, it does in fact apply
when the exponents are any real values.
Next, we have the quotient rule for
exponents, which tells us how to find the exponent when we are dividing two powers
of the same base. This time, we subtract the
powers. We can again demonstrate this rule
for positive integer values of 𝑚 and 𝑛 by writing a quotient out longhand and then
canceling common factors in the numerator and denominator. Here, we are assuming that 𝑚 is
greater than or equal to 𝑛, but the result is also true if 𝑛 is greater than
𝑚. Next, we have the power rule for
exponents, which tells us that if we’re raising a base to a power and then to
another power, overall, we’re raising that base to the product of those powers.
Those are the three main rules, but
there are also some additional ones that we need to recall. If we’re finding the power of a
product, this is the same as raising each factor to the power and then finding the
product. And if we’re finding the power of a
quotient, this is the same as raising both the numerator and denominator of that
quotient to the power separately. We should already be comfortable
applying each of these rules when the base is an integer or a fractional value. The focus of this video will be to
extend our knowledge to applying these rules to the entire set of real numbers. So we’re now including irrational
numbers as bases.
As we’re going to be working with
any real number here, we need to introduce some further rules for simplifying
expressions involving square roots. They are as follows. For an expression with a real base
𝑎, which must be nonnegative, the square root of 𝑎 squared is equal to 𝑎 and the
square root or 𝑎 squared is also equal to 𝑎. It is important that 𝑎 is
nonnegative here so that its square root is well defined in the first rule and so
that we have a unique answer when squaring and then square rooting in the
second. Let’s now consider our first
example, in which we’ll see how to use the product rule for exponents to simplify an
expression with a real base.
Fill in the blank. The expression root three
multiplied by root three squared multiplied by root three cubed simplifies to
blank.
We need to simplify this expression
which consists of three powers of the same base, root three, multiplied
together. We can therefore apply the product
rule for exponents, which states that for real values of 𝑎, 𝑚, and 𝑛, 𝑎 to the
power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚
plus 𝑛. Now there’s no exponent written for
the first term in the product, but we know that any number is equal to that number
to the first power. So root three is root three to the
power of one. So applying the product rule and
summing all three exponents, we have that this expression simplifies to root three
to the power of one plus two plus three, which is root three to the sixth power. We now know that we have six lots
of root three multiplied together, which we can write out longhand.
Next, we see that we can group the
terms in this product into pairs. And hence, we can express the
product as root three squared multiplied by root three squared multiplied by root
three squared. We can then recall the result that
for nonnegative values of 𝑎, the square root of 𝑎 squared is equal to 𝑎. And so the square root of three
squared is equal to three. The expression is therefore equal
to three multiplied by three multiplied by three or three cubed, which is equal to
27. So filling in the blank, the given
expression simplifies to 27.
Let’s now consider another example
in which we’ll see how to simplify the quotient of two powers in which the bases are
the negatives of one another.
What is the value of negative root
three to the sixth power divided by root three to the third power?
The bases in this expression are
nearly the same, but one is the negative of the other. We can’t yet apply laws of
exponents to simplify this division as the bases aren’t exactly the same. So let’s instead think about how we
can alter this expression so that the bases are identical.
In the first term, we have the
power of a product because negative root three is equal to negative one multiplied
by root three. We can therefore recall the power
of a product rule for exponents, which states that a product 𝑎𝑏 to the 𝑛th power
is equal to 𝑎 to the 𝑛th power multiplied by 𝑏 to the 𝑛th power. So we can rewrite the first term as
negative one to the sixth power multiplied by root three to the sixth power. We can then recall that negative
one to any even power is positive one. So in fact, the first term
simplifies to root three to the sixth power.
We now have exactly the same base
for both parts of the quotient. So we can simplify using the
quotient rule for exponents, which tells us that to divide powers of the same base,
we subtract the exponents. So root three to the sixth power
divided by root three to the third power is root three to the power of six minus
three, which is root three cubed. We can express this longhand as
root three multiplied by root three multiplied by root three. And then we can group the first two
terms in the product together to give root three squared multiplied by root
three.
We then recall that for nonnegative
real values of 𝑎, the square root of 𝑎 squared is equal to 𝑎. And so the square root of three
squared is equal to three. And the expression simplifies to
three root three. We found that the value of negative
root three to the sixth power divided by root three to the third power is three root
three.
Each of the examples we’ve
considered so far have involved only positive exponents. The rules, however, hold whether
the exponents are positive or negative. We need to recall some further key
properties to allow us to work with zero and negative exponents, both of which apply
when the base is nonzero. Firstly, any nonzero real base to
the power of zero is equal to one. Secondly, we recall that a negative
exponent defines a reciprocal. 𝑎 to the power of negative 𝑛 is
equal to one over 𝑎 to the power of 𝑛.
In our next example, we’re going to
apply the law for negative exponents to a problem in which the bases are
fractional. Before we do this, we can use the
second rule to write down a further property. This is called the power of a
quotient rule for negative exponents. And it states that for nonzero
values of 𝑎 and 𝑏, 𝑎 over 𝑏 to the negative 𝑛 is equal to 𝑏 over 𝑎 to the
power of 𝑛. We can briefly demonstrate why this
rule holds. Applying the law for negative
exponents to the expression 𝑎 over 𝑏 to the negative 𝑛 gives one over 𝑎 over 𝑏
to the 𝑛.
Applying the power of a quotient
rule in the denominator, this is equivalent to one over 𝑎 to the 𝑛th power over 𝑏
to the 𝑛th power. But we know that dividing by a
quotient is equivalent to multiplying by its reciprocal. So this is equivalent to one
multiplied by 𝑏 to the 𝑛th power over 𝑎 to the 𝑛th power. And then applying the power of a
quotient rule in reverse, this is equal to 𝑏 over 𝑎 to the 𝑛th power. Let’s now see how we can apply this
rule in an example.
For 𝑎 equal to 15 over four and 𝑏
equal to five over two, evaluate 𝑎 squared 𝑏 to the negative three.
Let’s begin by substituting the
numeric values of 𝑎 and 𝑏 into the given expression. And we have 15 over four squared
multiplied by five over two to the negative three. To simplify the second term in this
product, we can recall the power of a quotient rule for negative exponents. This states that for 𝑎 and 𝑏 not
equal to zero, 𝑎 over 𝑏 to the negative 𝑛 is equal to 𝑏 over 𝑎 to the power of
𝑛. So leaving the first term in the
product unchanged, the second term is equal to two-fifths to the power of three.
We then recall the power of the
quotient rule, which states that 𝑎 over 𝑏 to the 𝑛th power is equal to 𝑎 to the
𝑛th power over 𝑏 to the 𝑛th power. So applying this rule to each term,
we have 15 squared over four squared multiplied by two cubed over five cubed. We can then write the expression
out longhand and cancel common factors in the numerator and denominator of the
quotient. When we do, we’re left with three
multiplied by three in the numerator over two multiplied by five in the denominator,
which is equal to nine over 10. So we found that when 𝑎 is equal
to 15 over four and 𝑏 is equal to five over two, 𝑎 squared 𝑏 to the negative
three is equal to nine-tenths.
The final thing we’re going to
consider is how we can solve equations in which either the bases or the exponents
are the same. Suppose first we have the equation
two to the power of 𝑥 is equal to two to the power of four. As the bases on each side of the
equation are the same, the only way these two expressions can be equal is if the
exponents are also equal. So equating the exponents, we have
the solution 𝑥 is equal to four. In a similar way, suppose we have
the equation 𝑦 to the fourth power is equal to three to the fourth power.
As this time the exponents of the
two expressions are the same, we may think that the bases must be the same too. But in fact, it’s a little bit more
complicated than that because it depends on whether the power is even or odd. Three to the fourth power is equal
to 81, but so is negative three to the fourth power. So 𝑦 could be equal to three, but
it could also be equal to negative three. We would express this by saying the
absolute value of 𝑦 is equal to three. If however the exponent is an odd
integer, so suppose we have the equation 𝑧 cubed equals three cubed, then we don’t
have this same issue. Three cubed is equal to 27, but
negative three cubed is equal to negative 27. In this case, we can say that if
the exponents are the same, then the bases must also be the same.
We can formalize each of these
processes for solving equations using the following rules. Firstly, if 𝑎 to the 𝑚th power
equals 𝑎 to the 𝑛th power, where 𝑎 is a real number but not negative one, zero,
or one, then 𝑚 must be equal to 𝑛. And secondly, if 𝑎 to the 𝑛th
power is equal to 𝑏 to the 𝑛th power, then 𝑎 is equal to 𝑏 when 𝑛 is an odd
integer and the absolute value of 𝑎 is equal to the absolute value of 𝑏 when 𝑛 is
an even integer. In our final example, we’ll
demonstrate how we can use these laws to solve an equation in which the unknown is
in the exponent.
Find the value of 𝑥 in the
equation 16 multiplied by two to the power of nine minus 𝑥 is equal to two to the
power of 𝑥 minus one squared.
To find the value of 𝑥, we can use
laws of exponents. But in order to do this, we need to
first express every term in the equation as a power of the same base. We have two to the power of nine
minus 𝑥 and two to the power of 𝑥 minus one all squared. So let’s think about how we can
rewrite the term 16. Well, 16 is two to the fourth
power. So replacing 16 with two to the
fourth power, we have the entire equation in terms of the same base.
Now we can start to apply some laws
of exponents. On the left-hand side, we have a
product of powers of the same base. So we can apply the product rule
for exponents, which states that to multiply powers of the same base, we add the
exponents. The left-hand side is therefore
equivalent to two to the power of four plus nine minus 𝑥. On the right-hand side, we have a
power raised to another power. So we can simplify this using the
power rule of exponents, which states that in this instance, we multiply the powers
together. So the right-hand side becomes two
to the power of two multiplied by 𝑥 minus one. Simplifying the exponent on each
side, we have two to the power of 13 minus 𝑥 equals two to the power of two 𝑥
minus two.
Now as the bases on each side of
the equation are the same, it must also be true that the exponents are the same. So equating the exponents gives the
equation 13 minus 𝑥 equals two 𝑥 minus two. We now solve this equation for
𝑥. Adding both two and 𝑥 to each side
gives 15 equals three 𝑥. And then, dividing both sides of
the equation by three, we find that 𝑥 is equal to five. So the value of 𝑥 in the given
equation is five.
Let’s now summarize the key points
from this video. We can use the laws of exponents to
simplify powers with real bases. For real values of 𝑎 and 𝑏 and
integer values of 𝑚 and 𝑛, the product rule for exponents, the quotient rule for
exponents, the power rule for exponents, the power of a product rule, and the power
of a quotient rule all apply. We also introduced the power of a
quotient rule for negative exponents. For nonzero values of 𝑎 and 𝑏, 𝑎
over 𝑏 to the negative 𝑛 is equal to 𝑏 over 𝑎 to the power of 𝑛. We also saw some key rules for
working with squares and square roots. For nonnegative real values of 𝑎,
the square root of 𝑎 squared is equal to 𝑎 and the square root of 𝑎 squared is
also equal to 𝑎.
To solve simple exponential
equations with real bases, we can use the following rules when either the bases or
exponents are equal. If 𝑎 to the 𝑚 equals 𝑎 to the
𝑛, where 𝑎 is a real number but not negative one, zero, or one, then 𝑚 must be
equal to 𝑛. And if the exponents are the same,
so if 𝑎 to the 𝑛 equals 𝑏 to the 𝑛, then 𝑎 is equal to 𝑏 when 𝑛 is an odd
integer and the absolute value of 𝑎 is equal to the absolute value of 𝑏 when 𝑛 is
an even integer.
