Video Transcript
Simplify the expression negative
the square root of two raised to the power of negative three times root two squared
all raised to the power of negative three.
In this question, we are asked to
evaluate an expression involving operations applied to various exponential
expressions. To simplify this expression, we
will want to apply the laws of exponents. However, we can note that the bases
are not the same. So we will start by rewriting the
first factor to have a base of root two. We can rewrite this factor as
negative one times root two all raised to the power of negative three. We can then simplify this by
recalling that 𝑎 times 𝑏 all raised to the power of 𝑛 is equal to 𝑎 raised to
the power of 𝑛 times 𝑏 raised to the power of 𝑛.
Applying this result to the factor
gives us negative one raised to the power of negative three times root two raised to
the power of negative three times root two squared all raised to the power of
negative three. We can now see that we are
multiplying two exponential expressions with the same base. So we can simplify this expression
by using the product rule for exponents, which tells us 𝑎 raised to the power of 𝑚
times 𝑎 raised to the power of 𝑛 is equal to 𝑎 raised to the power of 𝑚 plus
𝑛. Applying this law gives us negative
one raised to the power of negative three times root two raised to the power of
negative three plus two all raised to the power of negative three.
We can then evaluate negative one
raised to the power of negative three to obtain negative one. And we can also evaluate the
expression in the exponent of root two to get root two raised to the power of
negative one. We now have a product raised to a
power. So we can once again use the laws
of exponents to raise each factor to the power. We have negative one raised to the
power of negative three times root two raised to the power of negative one raised to
the power of negative three.
We now want to evaluate the first
factor. Once again, we know that negative
one raised to the power of negative three is negative one. For the second factor, we can note
that we have a base raised to a power and then another power in turn. So we can apply the power rule for
exponents, which tells us that 𝑎 raised to the power of 𝑚 all raised to the power
of 𝑛 is equal to 𝑎 raised to the power of 𝑚 times 𝑛. Applying this result to the factor
gives us root two raised to the power of negative one times negative three. We can then evaluate the expression
in the exponent to obtain negative one times root two cubed.
We now want to evaluate this
expression. We can recall that cubing root two
is the same as multiplying the square of root two by root two, so this is two times
root two. Finally, we multiply this by
negative one to get negative two root two.
