Video Transcript
Simplify six to the power of
negative three multiplied by three all over two to the power of negative one.
To begin, it’s sensible to begin by
evaluating each of the numbers in this expression which have exponents. They are six to the power of
negative three and two to the power of negative one. But what does a negative power or a
negative exponent actually mean?
Well, a negative exponent tells us
to find the reciprocal of that number. So 𝑎 to the power of negative 𝑏
is equal to one over 𝑎 to the power of 𝑏. For example, two to the power of
negative three is equal to one over two to the power of three. And since two to the power of three
is eight, we can say that two to the power of negative three is the same as
one-eighth.
And it works in much a similar way
when we’re dealing with fractions. Four-thirds to the power of
negative two is the same as three-quarters squared. And this is because the reciprocal
of four-thirds is three-quarters. We can then evaluate the
numerator. Three squared is nine. And the denominator, four squared
is 16. So four-thirds all squared is the
same as nine 16ths.
Let’s use this to evaluate six to
the power of negative three. The reciprocal of six is
one-sixth. So six to the power of negative
three is one over six cubed. We could evaluate six cubed. But it’s probably not the most
sensible step to take. And we’ll see why in a moment. The reciprocal of two is
one-half. So two to the power of negative one
is one over two to the power of one. And since anything to the power of
one is just itself, it’s the same as one-half.
So let’s rewrite our expression
piece by piece. It’s one over six cubed multiplied
by three all over one-half. And next, we recall the fact that
this fraction line means divide. And to divide by a fraction, we
multiply by the reciprocal of that fraction. So this is the same as one over six
cubed multiplied by three multiplied by two over one.
Now multiplication is
commutative. It can be done in any order. So we don’t really need these
brackets here. And we add a denominator of one to
the number three. We then multiply the
numerators. One multiplied by three multiplied
by two is six. And six cubed multiplied by one
multiplied by one gives us a denominator of six cubed. Now remember, six is the same as
six to the power of one.
And this time, we recall this
rule. 𝑥 to the power of 𝑎 divided by 𝑥
to the power of 𝑏 is 𝑥 to the power of 𝑎 minus 𝑏. When we divide as long as the basis
that sees big numbers are the same, we can subtract the exponents. So, one minus three is negative
two. And we can see that six to the
power of one divided by six to the power of three is six to the power of negative
two.
And we’ve already seen that a
negative power tells us to find the reciprocal. So six to the power of negative two
is one over six squared. And since six squared is 36, we can
say that one over six squared is the same as one 36th. And we have fully simplified this
expression.
Six to the power of negative three
multiplied by three over two to the power of negative one is the same as one
36th.
