Video Transcript
Simplify negative 𝑥 over 𝑦 all
raised to the power of negative three all over negative 𝑥 over 𝑦 all raised to the
10th power.
In this question, we are asked to
simplify an expression involving the quotient of two bases raised to different
exponents. The first thing we want to notice
is that the base in both expressions are the same. Remember that the base is the part
of the expression that is being raised to a power. We can then note that this
expression is in the same form as the quotient rule for exponents. This tells us that 𝑏 to the power
of 𝑚 over 𝑏 to the power of 𝑛 is equal to 𝑏 to the power of 𝑚 minus 𝑛. We raise the base to the power of
the difference of the exponents.
We can directly apply this rule to
simplify the expression. Our base 𝑏 is negative 𝑥 over 𝑦,
our value of 𝑚 is negative three, and our value of 𝑛 is 10. We obtain negative 𝑥 over 𝑦 all
raised to the power of negative three minus 10. We can then evaluate the expression
in the exponent to obtain negative 𝑥 over 𝑦 to the power of negative 13.
We could stop here. However, it is worth noting that
there are many ways we can simplify this expression. For instance, we can recall that
raising a base to a negative exponent is the same as raising the reciprocal of the
base to the positive exponent. So we have that 𝑎 over 𝑏 all
raised to the power of negative 𝑛 is equal to 𝑏 over 𝑎 all raised to the power of
𝑛.
We can use this to rewrite our
answer. The reciprocal of the base is
negative 𝑦 over 𝑥, and we need to raise this base to the power of positive 13. We can rewrite this one more time
by recalling that raising a fraction to an exponent is the same as raising both the
numerator and denominator to that exponent separately. So, 𝑏 over 𝑎 all raised to the
power of 𝑛 is equal to 𝑏 to the power of 𝑛 over 𝑎 to the power of 𝑛. Applying this result with a
numerator of negative 𝑦 gives us negative 𝑦 to the 13th power over 𝑥 to the 13th
power.
We could even rewrite this further
by noting that raising a negative number to an odd exponent is the same as raising
the positive base to the exponent and multiplying by negative one. However, since there are so many
ways of rewriting this expression, we will leave our answer as negative 𝑥 over 𝑦
all raised to the power of negative 13.
