### Video Transcript

Which of the following expressions
has the same value as two over three raised to the power of negative three all
raised to the power of five? Is it option (A), two raised to the
15th power over three raised to the 15th power? Option (B), two raised to the
eighth power over three raised to the eighth power. Option (C), three raised to the
15th power divided by two raised to the 15th power. Option (D), three raised to the
eighth power over two raised to the eighth power. Or is it option (E), three squared
over two squared?

In this question, we are given an
expression involving raising a rational number to two different powers in turn and
asked to determine which of five given expressions has the same value as this
expression. To answer this question, we can
begin by looking at the form all of the answers are given in. We can note that every answer is of
the form 𝑎 raised to the 𝑛th power over 𝑏 raised to the 𝑛th power. Therefore, to answer this question,
we are going to want to rewrite our given expression so that the exponents are
applied to each of two and three separately.

There are many ways that we could
do this. We can start by noting that we are
raising a base to two exponents in turn. And so we can apply the power rule
for exponents, which states that raising a base to two exponents in turn is equal to
raising the base to the product of the exponents. So, 𝑥 raised to the power of 𝑚
all raised to the power of 𝑛 is equal to 𝑥 raised to the power of 𝑚 times 𝑛.

Applying this result with 𝑏 equal
to two-thirds, 𝑚 equal to negative three, and 𝑛 equal to fives yields two-thirds
raised to the power of negative three times five, which is equal to two-thirds
raised to the power of negative 15. We can simplify the expression
further by recalling that raising a nonzero base to a negative exponent is
equivalent to raising the reciprocal of the base to the positive exponent. So, 𝑥 over 𝑦 all raised to the
power of negative 𝑛 is equal to 𝑦 over 𝑥 raised to the power of 𝑛. This is provided that 𝑥 and 𝑦 are
nonzero. Applying this result to our
expression gives us three-halves raised to the 15th power.

Finally, we can recall that when
raising a fraction to an exponent, we can instead raise the numerator and
denominator to the exponent separately. So, 𝑦 over 𝑥 all raised to the
𝑛th power is equal to 𝑦 raised to the 𝑛th power divided by 𝑥 raised to the 𝑛th
power. Applying this result to our
expression gives us three raised to the 15th power over two raised to the 15th
power, which we can see matches option (C).