Video Transcript
Which of the following is equal to
negative three-quarters to the power of five multiplied by negative three-quarters
to the power of negative seven. The options are (A) one and
seven-ninths, (B) negative three-quarters to the power of negative 35, (C) negative
one and seven-ninths, (D) nine over 16, or (E) negative nine over 16.
So, in order to solve this problem,
what we’re gonna do is look at a couple of general rules we have for exponents. First of all, if we have 𝑥 to the
power of 𝑎 multiplied by 𝑥 to the power of 𝑏, this is equal to 𝑥 to the power of
𝑎 plus 𝑏. So, we add the exponents. And for the second rule, we’ve got
𝑥 over 𝑦 to the power of negative 𝑛 is equal to 𝑦 to the power of 𝑛 over 𝑥 to
the power of 𝑛. So, what we do is we find the
reciprocal of our fraction and then we put both the numerator and denominator to the
power of 𝑛.
And we can use our first rule
because we notice in our question that we, in fact, have both of our bases are the
same because they’re both negative three over four or negative three-quarters. So therefore, negative
three-quarters to the power of five multiplied by negative three-quarters to the
power of negative seven is gonna be equal to negative three-quarters to the power of
five add negative seven. Which is gonna be equal to negative
three-quarters to the power of negative two.
So, now, what we’re gonna do is
move on and apply the second rule. So, to apply the second rule, what
we do, first of all, is find the reciprocal. So, instead of three over four,
we’re gonna have four over three. But because we had negative three
over four, what we’re now gonna have is negative four over three. But what I’m gonna do is I’m gonna
put the negative with the numerator just so it doesn’t get left out or just because
we don’t get the wrong answer, i.e., negative when we should get positive, et
cetera.
So, then, we’ve got negative four
squared over three squared. And that’s because they’re both to
the power of two. And this is gonna give us 16 over
nine. And that’s because negative four
multiplied by negative four is 16 and three multiplied by three is nine. And then, to change this it into a
mixed number, what we do is we see how many nines go into 16. And they go into 16 once remainder
seven. And we do that because if we looked
at our answers (A), (B), (C), (D), and (E), none of them are 16 over nine.
So therefore, our answer is gonna
be one and seven-ninths. So, now, we can take a look at our
answers on the left-hand side. So therefore, we can match this to
answer (A). And we see that answer (A) must be
the correct answer. So, (A), which is one and
seven-ninths, is the correct answer. But also, it’s worth noting that we
could’ve got some of the other answers with some simple mistakes.
For example, we could’ve got answer
(C) if we’d forgotten about the negative sign that we’d mentioned earlier. Because if we’d forgotten about the
negative sign and just left it outside of our fraction, then what we would have had
is negative 16 over nine, which would have given us negative one and
seven-ninths.
And another common mistake that
could have brought us answer (B) would have been one where we multiplied the
exponents instead of adding them. So, we wouldn’t use the first
rule. We’d use the rule that was
incorrect that said if you multiply numbers that are the same base and different
exponents, then you multiply the exponents. And that would’ve given us negative
35. So, we’ve avoided that. And we’ve got the correct answer,
which is one and seven-ninths or answer (A).
