Video Transcript
Evaluate the following expression:
six raised to the power of negative two multiplied by two raised to the power of
negative two.
In this question, we are asked to
evaluate an expression involving the product of two different bases raised to
negative exponents. There are many ways that we can
evaluate this expression, and we will go through two of these.
First, we can recall that raising a
base to a negative exponent is equivalent to raising the reciprocal of the base to
the positive exponent. We have 𝑏 raised to the power of
negative 𝑛 is equal to one over 𝑏 raised to the power of 𝑛. Applying this result to each of the
factors separately gives us one over six squared times one over two squared.
We can then recall that squaring a
number means multiplying it by itself. So six squared is six times six,
which is equal to 36. And two squared is two times two,
which is equal to four. This gives us one over 36
multiplied by one over four. We can then multiply these
fractions by multiplying their numerators and denominators. We obtain one times one over 36
times four, which we can calculate is equal to one over 144.
A second method of answering this
question is to note that we are multiplying two bases raised to the same
exponent. This means that we can apply the
power of a product rule, which states that 𝑎 raised to the power of 𝑛 times 𝑏
raised to the power of 𝑛 is equal to 𝑎 times 𝑏 all raised to the power of 𝑛. Applying this result with 𝑎 equal
to six, 𝑏 equal to two, and 𝑛 equal to negative two gives us six times two all
raised to the power of negative two.
We can then calculate that six
times two is 12. So we have 12 raised to the power
of negative two. We can then note that this is a
base raised to a negative exponent. So we can take the reciprocal of
the base and raise it to the positive exponent to obtain one over 12 squared, which
we can calculate is also one divided by 144.
In either case, we have shown that
six raised to the power of negative two times two raised to the power of negative
two is equal to one divided by 144.
