Video Transcript
Given that 𝑧 equals negative
one-fourth, find the numerical value of negative eight 𝑧 to the fourth power times
𝑧 squared all over 16𝑧 cubed.
Starting with this expression, we
want to get it in its simplest form before we try and calculate its numerical
value. In simplifying, we want to get rid
of these parentheses. We remember that 𝑥 times 𝑦 to the
𝑎 power equals 𝑥 to the 𝑎 power 𝑦 to the 𝑎 power so that we have negative eight
to the fourth power times 𝑧 to the fourth power times 𝑧 squared. When we bring over the denominator
16𝑧 cubed, we can do some canceling between the numerator and the denominator. 𝑥 to the 𝑎 power over 𝑥 two to
the 𝑏 power equals 𝑥 to the 𝑎 minus 𝑏 power.
Since we have 𝑧 to the fourth
power in the numerator and 𝑧 cubed in the denominator, the 𝑧 cubed in the
denominator cancels out and the 𝑧 to the fourth power in the numerator becomes 𝑧
to the first power. Four minus three is one. There are no more 𝑧-variables in
the denominator, but in the numerator we have 𝑧 to the first power times 𝑧
squared. We can simplify that by writing it
as 𝑧 cubed; one plus two is three. Now, we have negative eight to the
fourth power times 𝑧 cubed over 16.
We can no longer simplify that
𝑧-variable. But it is worth considering if we
can simplify the negative eight to the fourth power over 16. I know that negative eight could
have a factor of negative four and two. If I write negative eight in this
way, we would then have negative four to the fourth power times two to the fourth
power times 𝑧 cubed over 16, which is helpful to ask because two to the fourth
power is 16. So, the factor of 16 in the
numerator and the factor of 16 in the denominator cancel out.
We now have negative four to the
fourth power times 𝑧 cubed. And we’re ready to plug in negative
one-fourth for 𝑧. Negative four to the fourth power
times negative one-fourth to the third power. If we have 𝑥 over 𝑦 to the 𝑎
power, that’s equal to 𝑥 to the 𝑎 power over 𝑦 to the 𝑎 power. If we break up this fraction, we’ll
have negative four to the fourth power times negative one cubed over four cubed. I wanna break up this negative four
to the fourth power one final time so that we have negative one to the fourth power
times four to the fourth power.
Negative one to the fourth power
equals one. And since we have four cubed in the
denominator and four to the fourth power in the numerator, we can say that we’ll
have four to the first power in the numerator. We subtract those exponents. Four minus three is one. Four to the first power equals
four. And the final operation we need to
do is negative one cubed. That’s negative one times negative
one, which is positive one, times negative one, which will be negative. So, our final answer here, the
numerical value for this expression is negative four.
