Video Transcript
Simplify the square root of 100𝑥
to the 16th power.
In this question, the value of 𝑛
in our 𝑛th root is omitted. When it is omitted, we assume that
it’s equal to two, and that’s why we define it as being the square root. So we’re going to simplify this
expression by applying some of the properties of 𝑛th roots, where 𝑛 is equal to
two. The first property we apply looks
at finding the product of 𝑛th roots. This tells us that when the 𝑛th
root of 𝑎 and the 𝑛th root of 𝑏 are well defined in real numbers, then the 𝑛th
root of 𝑎 times 𝑏 is also defined. It’s the 𝑛th root of 𝑎 times the
𝑛th root of 𝑏. We can omit this value of 𝑛 and
say that equivalently the square root of 𝑎 times the square root of 𝑏 is equal to
the square root of 𝑎𝑏.
We can then apply this property in
reverse. And it allows us to separate the
square root of 100𝑥 to the 16th power and write it as the square root of 100 times
the square root of 𝑥 to the 16th power. And of course, if we know our
square numbers by heart, the first part of this expression is quite easy to
evaluate. The square root of 100 is simply
equal to 10. But what do we do with the square
root of 𝑥 to the 16th power? Well, we’re going to use the rule
that applies when 𝑛 is an even integer and 𝑎 is a real number. This is that the 𝑛th root of 𝑎 to
the 𝑛th power is equal to the absolute value of 𝑎.
To be able to apply this rule, we
do need to do a little bit of manipulation, and that involves applying one of our
exponent laws. This says that 𝑥 to the eighth
power squared is equal to 𝑥 to the 16th power. So we can equivalently write the
square root of 𝑥 to the 16th power as the square root of 𝑥 to the eighth power
squared. And then, by the earlier property,
we can say that this is actually equal to the absolute value of 𝑥 to the eighth
power. We can now replace each part of our
earlier expression with the values 10 and the absolute value of 𝑥 to the eighth
power. And we see that we’ve simplified
the square root of 100𝑥 to the 16th power as 10 times the absolute value of 𝑥 to
the eighth power.
Then, let’s quickly think about
what the absolute value actually tells us. It takes the input, which is here
𝑥 to the eighth power, and it makes it positive. Of course, if 𝑥 is a real number,
then 𝑥 to the 𝑛th power will actually be nonnegative for even values of 𝑛. In this case, our power eight is
even, so we can say that 𝑥 to the eighth power will be nonnegative. It will be greater than or equal to
zero if 𝑥 is a real number. This means the absolute value
symbols are actually unnecessary here, and so we can further simplify our
expression. When we do, we find that the square
root of 100𝑥 to the 16th power can be simplified to 10 times 𝑥 to the eighth
power.
