Video Transcript
In this video, we will learn how to
simplify monomials with negative exponents. We will begin by recalling what we
know about positive exponents. We know that two to the fourth
power is equal to two multiplied by two multiplied by two multiplied by two. This is equal to 16. Two cubed or two to the third power
is equal to two multiplied by two multiplied by two, which is eight. In the same way, we know that two
squared is equal to four and two to the first power is two. When we have any expression 𝑥 to
the 𝑛th power where 𝑛 is a positive integer, then 𝑛 is telling us the number of
times we multiply 𝑥 by itself.
We can hopefully spot a pattern
where the exponent is decreasing by one. 16 divided by two is equal to
eight, eight divided by two is equal to four, and four divided by two is equal to
two. As the exponent decreases by one,
the number is divided by two. Following this pattern, we get two
to the power of zero is equal to one. We know this is true from our rule
of zero exponents, 𝑥 to the power of zero is equal to one. Let’s see what happens when we
repeat the pattern for negative exponents, two to the power of negative one, two to
the power of negative two, and two to the power of negative three.
We know that one divided by two is
equal to one-half. One-half divided by two is
one-quarter. And one-quarter divided by two is
one-eighth. We might recognize here that eight
is equal to two multiplied by two multiplied by two. And four is equal to two multiplied
by two. This means that two to the power of
negative three appears to be equal to one divided by two cubed. Likewise, two to the power of
negative two is equal to one over two squared. This leads us to the general rule
for negative exponents that, for any nonzero 𝑥, 𝑥 to the power of negative 𝑛 is
equal to one over 𝑥 to the power of 𝑛. We will now look at a simple
example involving simplifying exponents to prove this
Simplify two cubed divided by two
to the sixth power.
Two cubed or two to the third power
is equal to two multiplied by two multiplied by two. Two to the sixth power involves
multiplying six twos together. We can divide the numerator and
denominator by two three times. This leaves us with one over two
multiplied by two multiplied by two or one over two cubed. We know from our quotient rule of
exponents that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥
to the power of 𝑎 minus 𝑏. To simplify two cubed divided by
two to the sixth power, we can subtract the exponents. Three minus six is equal to
negative three. This means that two cubed divided
by two to the power of six is equal to two to the power of negative three or one
over two cubed. This confirms the rule for negative
exponents that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power
of 𝑛 or 𝑥 to the 𝑛th power.
In order to answer the remainder of
the questions in this video, we need to recall the five key exponent rules.
The product rule of exponents
states 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 add our exponents. The quotient rule of exponents
states that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to
the power of 𝑎 minus 𝑏. Thirdly, the power rule of
exponents states that 𝑥 to the power of 𝑎 raised to the power of 𝑏 is equal to 𝑥
to the power of 𝑎𝑏; we multiply the two exponents. When dealing with zero exponents,
we know that 𝑥 to the power of zero equals one. Finally, as just proved, with
negative exponents, 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the
power of 𝑛. We will now look at some questions
involving these rules.
Evaluate 14 to the power of
negative two.
We recall that our rule for
negative exponents states that 𝑥 to the power of negative 𝑛 is equal to one over
𝑥 to the power of 𝑛. This means that 14 to the power of
negative two is equal to one over 14 squared. 14 squared is equal to 14
multiplied by 14. There are many ways of calculating
this product. One way would be to multiply 14 by
10 and then 14 by four. 14 multiplied by 10 is 140 and 14
multiplied by four is 56. Adding these gives us 196. Therefore, 14 multiplied by 14 or
14 squared is 196. 14 to the power of negative two is
equal to one over 196.
Our next question involves the
quotient rule of exponents as well as negative exponents.
Evaluate two to the power of
negative two divided by two to the power of four.
We recall that our quotient rule of
exponents states that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is
equal to 𝑥 to the power of 𝑎 minus 𝑏. This means that two to the power of
negative two divided by two to the power of four or two to the fourth power is equal
to two to the power of negative two minus four. Negative two minus four is equal to
negative six. Our negative rule of exponents
states that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of
𝑛.
This means that two to the power of
negative six is equal to one over two to the sixth power. Two to the sixth power is the same
as six twos multiplied together. Two multiplied by two is equal to
four. Multiplying this by two gives us
eight. Eight multiplied by two is 16. 16 multiplied by two is 32. And 32 multiplied by two is 64. One over two to the sixth power is
the same as one over 64. The value of two to the power of
negative two divided by two to the power of four is one over 64.
Our next question involves
recognizing equivalent expressions.
Which of the following is
equivalent to three to the power of negative three? Is it (A) negative nine, (B)
negative one-ninth, (C) one-ninth, (D) one twenty-seventh, or (E) 27?
We know that the rule of negative
exponents states that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the
power of 𝑛. In this question, three to the
power of negative three is equal to one over three cubed. We know that three cubed is equal
to three multiplied by three multiplied by three. As three multiplied by three is
nine and nine multiplied by three is 27, then three cubed equals 27. Three to the power of negative
three is therefore equal to one over 27. The correct answer is option
(D).
We could have immediately ruled out
options (A) and (B) as there is no power that we can raise positive three to which
will result in a negative answer. In option (C), we’ve correctly used
the reciprocal. However, we’ve multiplied three by
three instead of three cubed on the denominator. Option (E) is three cubed and not
three to the power of negative three.
Our last two questions are more
complicated problems involving more than one of the rules of exponents.
Simplify five 𝑥 to the power of
negative eight squared multiplied by six 𝑥 squared squared.
We will begin this question by
simplifying each of the parentheses. We recall that 𝑥 to the power of
𝑎 all to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏. Five 𝑥 to the power of negative
eight squared can be rewritten as five squared multiplied by 𝑥 to the power of
negative eight squared. Five squared is equal to 25. Using the rule of exponents
mentioned, we need to multiply negative eight by two. This is equal to negative 16. So the first term simplifies to
25𝑥 to the power of negative 16.
We need to repeat this process for
the second part of our expression, six 𝑥 squared all squared. This is the same as six 𝑥 squared
multiplied by six 𝑥 squared, which can be written as six squared multiplied by 𝑥
squared squared. Six squared is 36. Multiplying the exponents once
again gives us 𝑥 to the fourth power or 𝑥 to the power of four. The original expression is
therefore simplified to 25𝑥 to the power of negative 16 multiplied by 36𝑥 to the
power of four.
The product rule of exponents
states 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 will use this to simplify 𝑥 to
the power of negative 16 multiplied by 𝑥 to the power of four. Negative 16 plus four is equal to
negative 12. We also need to multiply 25 by
36. This is equal to 900 as 25
multiplied by 30 is 750 and 25 multiplied by six is 150. Our final answer is 900𝑥 to the
power of negative 12.
We will complete this video by
looking at one more question.
Simplify 𝑥 to the eighth power to
the power of negative two multiplied by 𝑥 to the power of negative six to the power
of four divided by 𝑥 to the power of negative eight multiplied by 𝑥 to the power
of negative three.
We will begin this question by
simplifying the numerator and denominator separately. In order to do this, we recall
three of the rules of exponents. The product rule, 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 quotient rule, 𝑥 to the power
of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus
𝑏. And the power rule, 𝑥 to the power
of 𝑎 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏.
Let’s begin by using the power rule
to simplify 𝑥 to the power of eight to the power of negative two and 𝑥 to the
power of negative six to the power of four. In both of these cases, we need to
multiply the exponents or powers. Eight multiplied by negative two is
negative 16. Negative six multiplied by four is
equal to negative 24. So the numerator simplifies to 𝑥
to the power of negative 16 multiplied by 𝑥 to the power of negative 24. We can then use the product rule to
simplify the numerator and denominator. Negative 16 plus negative 24 is
equal to negative 40. So the numerator simplifies to 𝑥
to the power of negative 40. Negative eight plus negative three
is negative 11. So the denominator becomes 𝑥 to
the power of negative 11.
Our final step is to use the
quotient rule to simplify this expression. We need to subtract the exponents,
negative 40 minus negative 11. Subtracting negative 11 is the same
as adding 11. Negative 40 plus 11 is equal to
negative 29. Our expression simplifies to 𝑥 to
the power of negative 29. We could rewrite this expression
using our rule for negative exponents. 𝑥 to the power of negative 𝑛 is
equal to one over 𝑥 to the power of 𝑛. This means that 𝑥 to the power of
negative 29 is also equal to one over 𝑥 to the power of 29 or 𝑥 to the 29th
power.
We will now summarize the key
points from this video. Negative exponents are defined as
follows. For any nonzero 𝑥, we have that 𝑥
to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. When working with negative
exponents as in this video, we may need to use the following rules. We have the product rule of
exponents. 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 have the quotient rule. 𝑥 to the power of 𝑎 divided by 𝑥
to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. The power rule of exponents states
that 𝑥 to the power of 𝑎 raised to the power of 𝑏 is equal to 𝑥 to the power of
𝑎𝑏. Finally, when dealing with zero
exponents, 𝑥 to the power of zero is equal to one for any nonzero 𝑥.
