### Video Transcript

In this video, we will learn how to
perform operations and simplifications on expressions that involve integer
exponents. To help us understand how to do
this, we will begin by recalling how to evaluate a power.

We know that for a power with base
π, where π is any real number apart from zero, and exponent π, where π is a
positive integer, then π to the πth power is equal to π multiplied by π
multiplied by π and so on, where there are π πβs. For example, two to the fourth
power is equal to two multiplied by two multiplied by two multiplied by two, which
is equal to 16.

Next, we recall the rules for zero
and negative exponents. If the base π is any real number
apart from zero, then π to the power of zero is equal to one. When dealing with negative
exponents, π to the power of negative π is equal to one over π to the πth power,
where π is any real number apart from zero. In this video, we will only
consider integer values of π. However, it is important to note
that this holds for all values of π. Likewise, π to the πth power is
equal to one over π to the power of negative π. In our first example, we will use
these definitions to simplify a numerical expression with integer exponents.

Calculate 10 to the fifth power
multiplied by 25 squared multiplied by four divided by 20 squared multiplied by five
to the fifth power.

To calculate this expression, we
can write the powers in expanded form and then cancel common factors in the
numerator and denominator. The first term in the numerator, 10
to the fifth power, can be rewritten as 10 multiplied by 10 multiplied by 10
multiplied by 10 multiplied by 10. 25 squared can be written as 25
multiplied by 25. This means that the entire
numerator can be rewritten as shown.

We can do the same with the
denominator and rewrite 20 squared and five to the fifth power. Whilst we could cancel at this
stage, we can simplify our expression further by writing the numerator and
denominator as products of prime factors. We know that 10 is equal to two
multiplied by five. 25 can be rewritten as five
multiplied by five. We can rewrite four as two
multiplied by two. And 20 written as a product of
prime factors is two multiplied by two multiplied by five.

We can use the first three of these
to rewrite the numerator as shown. And repeating this process with the
denominator, we have the following expression. Next, we can cancel common
factors. Two multiplied by two multiplied by
two multiplied by two can be canceled from the numerator and denominator. We can also cancel seven fives from
the numerator and denominator. Our expression therefore simplifies
to two multiplied by five multiplied by five multiplied by two multiplied by two all
divided by one. This is equal to two cubed
multiplied by five squared. And since two cubed is equal to
eight and five squared is 25, we have eight multiplied by 25, which is equal to
200. 10 to the fifth power multiplied by
25 squared multiplied by four divided by 20 squared multiplied by five to the fifth
power is 200.

Sometimes when simplifying
expressions with integer exponents, we need to use the laws of exponents to do
this. Before looking at our next example,
we will recall these laws of exponents. In this video, we will only
consider the following laws when we have nonzero bases and integer exponents. Firstly, we have the product rule,
which states that π to the power of π multiplied by π to the power of π is equal
to π to the power of π plus π. Next, we have the quotient
rule. This states that π to the power of
π divided by π to the power of π is equal to π to the power of π minus π. The power of a product rule states
that ππ to the power of π is equal to π to the power of π multiplied by π to
the power of π. In a similar way, the power of a
quotient rule states that π over π to the power of π is equal to π to the power
of π divided by π to the power of π. Finally, we have the power rule,
which states that π to the power of π raised to the power of π is equal to π to
the power of π multiplied by π.

We will now consider a couple of
examples where we need to use these laws of exponents.

Simplify four to the power of π₯
multiplied by four to the power of four π₯ divided by four to the power of three π₯
minus four multiplied by 16 to the power of π₯.

In order to simplify this
expression, we will need to use the laws of exponents. We recall that in order to use
these, each term must have the same base. In this question, three of the four
terms have a base of four. Our first step is therefore to
rewrite 16 to the power of π₯ with a base of four. We know that four squared is equal
to 16. This means that we can rewrite 16
to the power of π₯ as four squared to the power of π₯. The initial expression can
therefore be rewritten as four to the power of π₯ multiplied by four to the power of
four π₯ divided by four to the power of three π₯ minus four multiplied by four
squared to the power of π₯.

We will now consider three of the
laws of exponents. Firstly, the product rule states
that π to the power of π multiplied by π to the power of π is equal to π to the
power of π plus π. This means that the numerator can
be rewritten as four to the power of π₯ plus four π₯. Before applying the same rule to
the denominator, we need to consider the power rule of exponents. This states that π to the power of
π all raised to the power of π is equal to π to the power of π multiplied by
π. As such, four squared raised to the
power of π₯ can be rewritten as four to the power of two π₯.

We can then use the product rule to
rewrite the denominator as four to the power of three π₯ minus four plus two π₯. π₯ plus four π₯ is equal to five
π₯, and three π₯ minus four plus two π₯ is equal to five π₯ minus four. So our expression simplifies to
four to the power of five π₯ divided by four to the power of five π₯ minus four.

We can now use the quotient rule,
which states that π to the power of π divided by π to the power of π is equal to
π to the power of π minus π. Subtracting five π₯ minus four from
five π₯, we have four to the power of five π₯ minus five π₯ minus four. Distributing the parentheses, the
exponent becomes five π₯ minus five π₯ plus four. And our expression simplifies to
four to the fourth power. Evaluating this, we get four
multiplied by four multiplied by four multiplied by four, which is equal to 256. Four to the power of π₯ multiplied
by four to the power of four π₯ divided by four to the power of three π₯ minus four
multiplied by 16 to the power of π₯ is equal to 256.

In our next example, we will use
the laws of exponents to simplify an algebraic expression with two different
bases.

Simplify π₯ to the power of π plus
nine multiplied by π¦ to the power of π plus four divided by π₯ to the power of π
plus three multiplied by π¦ to the power of π.

In order to simplify this
expression, we can use the laws of exponents. However, we notice that we have two
different bases. Since the laws only apply to
exponents with the same bases, we can rewrite the expression as shown. We have π₯ to the power of π plus
nine divided by π₯ to the power of π plus three multiplied by π¦ to the power of π
plus four divided by π¦ to the power of π. Recalling that the quotient rule
for exponents states that π to the power of π divided by π to the power of π is
equal to π to the power of π minus π, we can rewrite the first part of our
expression as π₯ to the power of π plus nine minus π plus three.

In the same way, the second part of
our expression simplifies to π¦ to the power of π plus four minus π. In both parts, the πβs cancel. And since nine minus three is equal
to six, we have π₯ to the power of six multiplied by π¦ to the power of four. The expression π₯ to the power of
π plus nine multiplied by π¦ to the power of π plus four divided by π₯ to the
power of π plus three multiplied by π¦ to the power of π written in its simplest
form is π₯ to the sixth power π¦ to the fourth power.

We will now look at one final
example where we need to simplify an algebraic expression using the laws of
exponents and by taking out common factors that are powers.

Find the value of π for which two
to the power of π₯ plus six minus two to the power of π₯ plus two is equal to π
multiplied by two to the power of π₯.

In order to find the value of π in
this equation, we need the left-hand side of the equation to be in the form of the
right-hand side. In other words, we need to rewrite
two to the power of π₯ plus six minus two to the power of π₯ plus two so it is in
the form π multiplied by two to the power of π₯.

We recall that the product rule of
exponents states that π to the power of π multiplied by π to the power of π is
equal to π to the power of π plus π. This means that we can rewrite the
first term as two to the power of π₯ multiplied by two to the power of six. Likewise, two to the power of π₯
plus two can be rewritten as two to the power of π₯ multiplied by two squared. Both of our terms now have a common
factor of two to the power of π₯.

Factoring this out, we have two to
the power of π₯ multiplied by two to the sixth power minus two squared. We know that two squared is equal
to four and two to the sixth power is 64. As such, our expression simplifies
to two to the power of π₯ multiplied by 60 or 60 multiplied by two to the power of
π₯. Since this is equal to π
multiplied by two to the power of π₯, we can compare the coefficients, giving us a
value of π equal to 60.

We will now finish this video by
summarizing the key points. We saw in this video that we can
simplify exponential expressions by calculating the value of powers and canceling
down common factors. We can also simplify exponential
expressions using the laws of exponents. We have the product rule for
exponents, which states that π to the power of π multiplied by π to the power of
π is equal to π to the power of π plus π. The quotient rule states that π to
the power of π divided by π to the power of π is equal to π to the power of π
minus π. Thirdly, the power rule of
exponents states that π to the power of π raised to the power of π is equal to π
to the power of π multiplied by π. We also have the power of a product
rule, which states that ππ to the power of π is equal to π to the power of π
multiplied by π to the power of π, and the power of a quotient rule, which states
that π over π to the power of π is equal to π to the power of π divided by π
to the power of π.

In this video, we only considered
these for nonzero bases and integer exponents. However, itβs important to note
that they also hold for noninteger exponents. We also saw in our last example
that we can use the laws of exponents to solve equations.