Video Transcript
In this video, we will learn how to
use the rules of negative and fractional exponents or indices to simplify numerical
expressions. We will begin by looking at the
rules we can use when dealing with negative exponents and fractional exponents.
When dealing with negative
exponents, we use the following rule: 𝑥 to the power of negative 𝑛 is equal to one
over 𝑥 to the power of 𝑛. Let’s consider a couple of
examples: firstly, five to the power of negative two and then two to the power of
negative four. Using our rule, five to the power
of negative two can be rewritten as one over five to the power of two or five
squared. As five squared is equal to 25,
five to the power of negative two is one over 25 or one twenty-fifth. In the same way, two to the power
of negative four is equal to one over two to the fourth power or two to the power of
four. This is the same as two multiplied
by two multiplied by two multiplied by two. Two to the power of four is equal
to 16; therefore, two to the power of negative four is one over 16.
Let’s now consider what happens
when we want to raise a fraction to a negative power. We know that one over a number is
known as the reciprocal of that number. When finding the reciprocal of a
fraction, we swap the numerator and denominator. This means that 𝑎 over 𝑏 to the
power of negative 𝑛 is equal to 𝑏 over 𝑎 to the power of 𝑛. In our example, four over three to
the power of negative two is the same as three over four to the power of two all
squared. When raising any fraction to a
power, we can raise the numerator and denominator to this power separately. Three-quarters or three over four
squared is the same as three squared over four squared. This is equal to nine over 16 or
nine sixteenths.
We will use these two rules when
dealing with negative exponents of integers and fractions. When dealing with fractional
exponents, we use the rule that 𝑥 to the power of one over 𝑛 is equal to the 𝑛th
root of 𝑥. This means that eight to the power
of one-third is the same as the cube root of eight. As two cubed is equal to eight, the
cube root of eight is equal to two. This means that eight to the power
of one-third equals two. When trying to calculate 𝑥 to the
power of 𝑚 over 𝑛, we begin by finding the 𝑛th root of 𝑥 and then raise this
answer to the power of 𝑚.
Let’s consider 16 to the power of
three over two. The denominator is two, so this is
the root. We need to take the square root of
16. As the numerator is three, we will
then cube this answer. The square root of 16 is four and
four cubed is equal to 64. Therefore, 16 to the power of three
over two is equal to 64.
We could also do these two
operations in the opposite order. We could raise 𝑥 to the power of
𝑚 and then take the 𝑛th root. In our example, we would calculate
16 cubed and then square root the answer. Whilst this gives us the same
answer, it is far more complicated to cube 16 and then square root 4096 than square
root 16 and then cube four. In the vast majority of questions,
it will be easier to find the root first.
Let’s now consider what happens
when we have a negative fractional exponent, for example, 32 to the power of
negative two-fifths. We recall that when we had a
negative exponent, we looked at the reciprocal. 32 to the power of negative
two-fifths is the same as one over 32 to the power of two-fifths. Our next step is to find the fifth
root of 32 and then square the answer. The fifth root of 32 is equal to
two. So, we are left with one over two
squared. 32 to the power of negative
two-fifths is, therefore, equal to one-quarter. 𝑥 to the power of negative 𝑚 over
𝑛 is equal to one over the 𝑛th root of 𝑥 raised to the power of 𝑚. We can use these three rules when
solving problems with fractional exponents.
We will now look at some questions
involving negative and fractional exponents.
Which of the following is equal to
four multiplied by the square root of six to the power of negative one? Is it (A) 24, (B) four root six,
(C) two root six over three, or (D) two-thirds?
When dealing with negative
exponents, we use the rule that 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 one is equal to one over 𝑥 to the power of one which is the same as one
over 𝑥. The square root of six to the power
of negative one is equal to one over the square root of six. This means that when we multiply
this by four, we get four over the square root of six or four over root six. As we have a radical or surd on the
denominator, we need to rationalize this fraction. We do so by multiplying the
numerator and denominator by the square root of six.
This gives us four root six over
six as multiplying the square root of six by the square root of six is six. This comes from the fact that one
of our laws of indices states that the square root of 𝑥 multiplied by the square
root of 𝑥 is equal to 𝑥. We can simplify this fraction by
dividing the numerator and denominated by two. This leaves us with two root six
over three. The correct answer is, therefore,
option (C). Four multiplied by the square root
of six raised to the power of negative one is equal to two root six over three.
Our next question involves a
fractional power that is negative.
Evaluate 64 over 27 to the power of
negative two-thirds.
In order to answer this question,
we need to use our knowledge of fractional and negative exponents or indices. Firstly, when we raise a fraction
to a negative power, this is the same as the reciprocal of the fraction to the
positive of the power. When finding the reciprocal of a
fraction, the numerator and denominator swap. This means that 64 over 27 to the
power of negative two-thirds is the same as 27 over 64 to the power of
two-thirds. We also recall that when raising a
fraction to a power, we can raise the numerator and denominator to the power
separately. This means that we have 27 to the
power of two-thirds over 64 to the power of two-thirds.
When raising any number to the
power of 𝑚 over 𝑛, we firstly find the 𝑛th root of the number and then raise this
answer to the power of 𝑚. To calculate 27 to the power of
two-thirds, we find the cube root of 27 and then square our answer. The cube root of 27 is equal to
three. As three squared is equal to nine,
27 to the power of two-thirds is equal to nine. We repeat this for the
denominator. The cube root of 64 is equal to
four. As four squared is equal to 16, 64
to the power of two-thirds is equal to 16. This means that 64 over 27 to the
power of negative two-thirds is equal to nine over 16 or nine sixteenths.
In our next question, we will need
to convert decimals into fractions first.
Evaluate 0.03125 to the power of
negative 0.2.
In this question, our first step
will be to convert the base number 0.03125 and our exponent negative 0.2 into
fractions. 0.03125 is equal to the fraction
3125 over 100000. This is because the five is in the
hundred thousandths column. It is not immediately obvious what
the highest common factor of 3125 and 100000 is. We could begin by dividing the
numerator and denominator by five, 25, 125, or any other factor of the numerator and
denominator. 3125 does divide into 100000 32
times. This means that the fraction in its
simplest form is one over 32. Our exponent in this question was
negative 0.2, and this is the same as negative two-tenths. Dividing the numerator and
denominator by two, this simplifies to negative one-fifth. We can, therefore, rewrite our
calculation as one over 32 raised to the power of negative one-fifth.
We recall that raising the fraction
𝑎 over 𝑏 to the power of negative 𝑛 is the same as raising the fraction 𝑏 over
𝑎 to the power of 𝑛. This means we can rewrite our
calculation as 32 over one to the power of one-fifth. Any integer divided by one is equal
to the integer. Therefore, this could be rewritten
as 32 to the power of one-fifth. Raising any number to the unit
fraction one over 𝑛 is the same as finding the 𝑛th root of the number. We need to calculate the fifth root
of 32. As two to the power of five is
equal to 32, the fifth root of 32 is two. 0.03125 raised to the power of
negative 0.2 is equal to two.
In our final question, we will
combine all of the skills we have used in this video.
Evaluate four to the power of three
over two multiplied by 32 to the power of negative 0.2 multiplied by two to the
power of negative three all divided by 32 to the power of negative 0.4 multiplied by
two to the power of negative two.
One way of approaching this problem
would be to work out each of the five terms individually. We could work out four to the power
of three over two, 32 to the power of negative 0.2, and so on. Whilst this would give us the
correct answer, there is a quicker method if we recognize that four and 32 are
powers of two. Four is equal to two squared and 32
is equal to two to the power of five. This means that we could rewrite
our calculation as shown. Four to the power of three over two
is the same as two squared to the power of three over two. 32 to the power of negative 0.2 is
equal to two to the power of five to the power of negative 0.2. And 32 to the power of negative 0.4
is equal to two to the power of five to the power of negative 0.4.
We can now use our laws of
exponents or indices. 𝑥 to the power of 𝑎 raised to the
power of 𝑏 is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. We can multiply two by three over
two, five by negative 0.2, and five by negative 0.4. Two multiplied by three over two is
equal to three. This means that the first term is
equal to two cubed. Five multiplied by negative 0.2 is
equal to negative one. Our second term is two to the power
of negative one. The numerator is completed by
multiplying by two to the power of negative three. Five multiplied by negative 0.4 is
equal to negative two. This means that the numerator
becomes two to the power of negative two multiplied by two to the power of negative
two.
Another one of our laws 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 need to add three negative one
and negative three on the numerator. This gives us negative one. So, we have two to the power of
negative one. We repeat this on the
denominator. Negative two plus negative two is
equal to negative four. Our equation becomes two to the
power of negative one over two to the power of negative four.
Our third 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. We need to subtract negative four
from negative one. This is the same as adding four to
negative one, which gives us three. This leaves us with two to the
power of three or two cubed. Two cubed is equal to eight. Therefore, this is the answer to
our calculation.
Had we tried to solve the problem
by working out each term individually, we could have used the rules that 𝑥 to the
power of 𝑎 over 𝑏 is equal to the 𝑏th root of 𝑥 raised to the power of 𝑎 and 𝑥
to the power of negative 𝑎 is equal to one over 𝑥 to the power of 𝑎. Four to the power of three over two
is equal to eight. 32 to the power of negative 0.2 or
negative one-fifth is equal to one-half. This means that 32 to the power of
negative 0.4 or negative two-fifths is equal to one-quarter. Two to the power of negative three
is one-eighth and two to the power of negative two is one-quarter.
This would have left us with eight
multiplied by a half multiplied by an eighth divided by a quarter multiplied by a
quarter. The numerator simplifies to
one-half and the denominator one sixteenth. As dividing a fraction by a
fraction is the same as multiplying the first fraction by the reciprocal of the
second, one-half divided by one sixteenth is also equal to eight. This confirms the answer we got
using our first method.
We will now summarize the key
points from this video. In this video, we recalled our
three laws of exponents where we need to add, subtract, and multiply powers. When dealing with negative
exponents, we saw that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the
power of 𝑛. We also learnt a rule for
fractional exponents and used a combination of these to answer problems involving
integers, fractions, and decimals.
