Video Transcript
Write the square root of 25𝑎
squared 𝑏 to the sixth power in its simplest form.
In order to simplify this
expression, we’re going to recall that if the 𝑛th root of 𝑎, the 𝑛th root of 𝑏,
and the 𝑛th root of 𝑐 are well defined for positive integers 𝑛, then their
product is the 𝑛th root of 𝑎𝑏𝑐. We’re going to apply this property
in reverse to allow us to write the original expression as the square root of 25
times the square root of 𝑎 squared times the square root of 𝑏 to the sixth
power. Now, we know that the square root
of 25 is simply equal to five. But what do we do with the square
root of 𝑎 squared and the square root of 𝑏 to the sixth power?
Well, since we’re finding the
square root, this is an 𝑛th root where 𝑛 is even; it’s two. So we use the following
property. If 𝑛 is even and 𝑎 is a real
number, then the 𝑛th root of 𝑎 to the 𝑛th power is equal to the absolute value of
𝑎. So we can actually rewrite the
square root of 𝑎 squared as the absolute value of 𝑎. To repeat this process for the
square root of 𝑏 to the sixth power, we need to rewrite 𝑏 to the sixth power as 𝑏
cubed squared, meaning that the square root of 𝑏 cubed all squared is equal to the
absolute value of 𝑏 cubed.
We can now replace each root with
the value we found. So the square root of 25 times the
square root of 𝑎 squared times the square root of 𝑏 to the sixth power is five
times the absolute value of 𝑎 times the absolute value of 𝑏 cubed. Then, since five is by its very
nature nonnegative, we can use the rules for multiplying absolute value expressions
to rewrite this as the absolute value of five 𝑎𝑏 cubed. And so in its simplest form, the
square root of 25𝑎 squared 𝑏 to the sixth power is equal to the absolute value of
five 𝑎𝑏 cubed.
