Video Transcript
Which of the following is equal to
negative three-quarters all raised to the power of negative two? Option (A) 16 over nine. Option (B) negative 16 over
nine. Option (C) negative one times
negative six over negative eight. Option (D) nine over 16. Or is it option (E) negative nine
over 16?
In this question, we are given an
expression involving a fraction raised to a negative exponent. And we are asked to determine which
of five given expressions is equal to the given expression. To do this, we can start by noting
that the exponent of the expression is negative. We can evaluate expressions with
negative exponents by recalling that a negative exponent is the same as the positive
exponent where we also take the reciprocal of the base. So, 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 apply this to the given
expression.
First, we can rewrite the base of
the expression to get negative three over four all raised to the power of negative
two. Next, we take the reciprocal of the
base so that we can change the power to be positive two. This is the same as setting 𝑛
equals two, 𝑎 equals negative three, and 𝑏 equals four in our exponent rule. We obtain four over negative three
all squared. We can simplify this expression
further by recalling that raising a fraction to an exponent is equivalent to raising
its numerator and denominator separately to that exponent.
So, 𝑏 over 𝑎 all raised to the
power of 𝑛 is equal to 𝑏 raised to the 𝑛th power over 𝑎 raised to the 𝑛th
power. We can use this to rewrite our
expression. We want to square the numerator and
denominator separately, so we have four squared over negative three squared. We can now evaluate this expression
by recalling that squaring a number means multiplying it by itself. So, 𝑎 squared is 𝑎 times 𝑎.
Applying this result to the
numerator and denominator gives us four times four over negative three times
negative three. We can calculate that four times
four is 16 and negative three times negative three is nine, since a negative times a
negative is a positive. This gives us 16 over nine, which
we can see matches option (A).
