Video Transcript
Simplify the square root of two to
the eight 𝑛 plus 10 power times 30 to the eight 𝑛 plus three power all over the
square root of two to the eight 𝑛 power times six to the eight 𝑛 power times five
to the eight 𝑛 plus four power.
As we copy down this problem, I’m
going to write the square root of two down as two to the one-half power since those
values are equal. So, we’ll have two to the one-half
power to the eight 𝑛 plus 10 power. So, the only change we made when
copying everything down was to rewrite the square root of two as two to the one-half
power.
To simplify this expression, one
rule will be really important. And that’s 𝑥 to the 𝑎 plus 𝑏
power is equal to 𝑥 to the 𝑎 power times 𝑥 to the 𝑏 power. This means that two to the one-half
power to the 𝑎𝑛 plus 10 power can be written as two to the one-half power to the
eight 𝑛 times two to the one-half power to the 10th power. We’ll do the same thing for 30 to
the eight 𝑛 plus three power. We’re breaking these up so that we
can work on simplifying.
The 30 to the eight 𝑛 plus three
power becomes 30 to the eight 𝑛 power times 30 cubed. We’ll just bring across two to the
one-half power to the eight 𝑛 power in the denominator and six to the eight 𝑛. And we’ll break up five to the
eight 𝑛 plus four power into five to the eight 𝑛 power times five to the fourth
power. At this point, we have two to the
one-half power to the eight 𝑛 power in the numerator and the denominator. And so, they cancel out.
It might seem like there’s nothing
else we can cancel out. However, we have six to the eight
𝑛 power and five to the eight 𝑛 power. Six times five is 30, and so we can
rewrite six to the eight 𝑛 power times five to the eight 𝑛 power as 30 to the
eight 𝑛 power. Then, bring over the five to the
fourth power. And from there, we’ll need to note
that 𝑥 to the 𝑎 power to the 𝑏 power can be rewritten as 𝑥 to the 𝑎 times 𝑏
power. And that means two to the one-half
power to the 10th power can be rewritten as two to the one-half times 10 power,
which is two to the fifth power.
Now, we have two to the fifth power
times 30 to the eight 𝑛 power times 30 cubed all over 30 to the eight 𝑛 power
times five to the fourth power. 30 to the eight 𝑛 power in the
numerator and the denominator cancels out. And now we have two to the fifth
power times 30 cubed over five to the fourth power. There’s one final thing we can
simplify. If we remember that 30 is five
times six, we can write 30 cubed as five times six cubed, which is equal to five
cubed times six cubed.
If we bring everything else over,
we see that we have five cubed in the numerator and five to the fourth power in the
denominator. This means that the five cubed in
the numerator can be cancelled out, and five to the fourth power in the denominator
becomes five to the first power. So, we have six cubed times two to
the fifth power over five. Six cubed equals 216, two to the
fifth power is 32, all over five. And there’s nothing else that we
can simplify. So, we just multiply 216 by 32,
which gives us 6912 over five.
