Video Transcript
Simplify five to the seventh power
plus the square root of five to the 14th power plus the square root of five to the
14th power plus the square root of five to the 14th power plus the square root of
five to the 14th power.
The first thing that I wanna do is
rewrite all of these square roots as five to the one-half power. The square root of five is the same
thing as saying five to the one-half power. And we’re still taking this five to
the one-half power to the 14th power in all four cases.
And then, we’ll need to consider
the power rule. 𝑥 to the 𝑎 power to the 𝑏 power
is the same thing as saying 𝑥 to the 𝑎 times 𝑏 power. And that means five to the one-half
power to the 14th power equals five to the one-half times 14 power. And that would be true in all four
of these cases.
We can multiply 14 times one-half,
which equals seven. We’re dealing with five to the
seventh power. All four of these simplify to five
to the seventh power. And we can’t forget the other five
to the seventh power term we started with.
What’s happening now is we’re
adding five to the seventh power to itself five times. Let’s consider what 𝑥 to the 𝑎
power plus 𝑥 to the 𝑎 power would be. It would be two times 𝑥 to the 𝑎
power. We have two 𝑥 to the 𝑎 power
terms. In our case, we have five five to
the seventh power terms, five times five to the seventh power, which we can rewrite
like this: five times five to the seventh power.
We’ll need another rule. This is for multiplying exponents
with the same base and different powers. 𝑥 to the 𝑎 power times 𝑥 to the
𝑏 power equals 𝑥 to the 𝑎 plus 𝑏 power. And we have the whole number five,
which we could write as five to the first power.
In that case, we would add one plus
seven. We would add the exponents and find
that this expression simplifies to five to the eighth power.
